Systems of Linear Equations Alıştırma Soruları — SAT

Consider the system of linear equations below.

\begin{aligned} 3x - 4y &= 7 \\ 2x + 3y &= 16 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x + y$ ?

7

6

14

-9

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

The correct answer is 7. By multiplying the first equation by 3 and the second by 4, we get

9x - 12y = 21

and

8x + 12y = 64

. Adding these equations eliminates the

y

terms and gives

17x = 85

, which simplifies to

x = 5

. Substituting

x = 5

into the second equation yields

2(5) + 3y = 16 \implies 10 + 3y = 16 \implies 3y = 6 \implies y = 2

. The sum of the coordinates is

x + y = 5 + 2 = 7

.

Adım Adım Çözüm

1

Multiply the equations by constants to align the coefficients of one variable for elimination.

Multiplying the first equation by 3 gives

9x - 12y = 21

. Multiplying the second equation by 4 gives

8x + 12y = 64

.

This sets up the system so that the

y

terms have opposite coefficients (

-12

and

+12

), allowing them to be eliminated by addition.

2

Add the two modified equations to eliminate the

y

variable and solve for

x

.

(9x - 12y) + (8x + 12y) = 21 + 64 \implies 17x = 85 \implies x = 5

.

Adding the equations eliminates

y

and leaves a single linear equation in terms of

x

.

3

Substitute the value of

x

back into one of the original equations to solve for

y

.

Substituting

x = 5

into the second equation

2x + 3y = 16

gives

2(5) + 3y = 16 \implies 10 + 3y = 16 \implies 3y = 6 \implies y = 2

.

Plugging the known variable value back in allows us to solve for the remaining unknown variable.

4

Calculate the value of the requested expression

x + y

.

x + y = 5 + 2 = 7

.

The question specifically asks for the sum of

x

and

y

.

Anahtar Kavram

Solving systems of linear equations using the elimination method and evaluating linear combinations of the solutions.

Alternatif Yöntem

Alternatively, you can solve the first equation for

x

in terms of

y

:

3x = 7 + 4y \implies x = \frac{7+4y}{3}

. Substitute this expression into the second equation:

2\left(\frac{7+4y}{3}\right) + 3y = 16

. Multiply the entire equation by 3 to clear the fraction:

2(7+4y) + 9y = 48 \implies 14 + 8y + 9y = 48 \implies 17y = 34 \implies y = 2

. Substitute

y = 2

back to find

x

:

x = \frac{7+4(2)}{3} = 5

. Thus,

x + y = 5 + 2 = 7

.

Tahmini Süre:1m 30s

Soru 2Soru →

In the system of equations below, what is the value of $y$ ?

4x + 3y = 25

2x + 3y = 17

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

y

is

3

.

Subtracting the second equation from the first equation gives

(4x + 3y) - (2x + 3y) = 25 - 17

, which simplifies to

2x = 8

. Dividing by 2 gives

x = 4

. Substituting

x = 4

into the second equation gives

2(4) + 3y = 17

, which simplifies to

8 + 3y = 17

. Subtracting 8 from both sides gives

3y = 9

, and dividing by 3 gives

y = 3

.

Adım Adım Çözüm

1

Subtract the second equation from the first equation to eliminate the

y

term.

2x = 8

Subtracting the equations eliminates

3y

since it is common to both equations, leaving a single variable equation.

2

Solve for

x

by dividing both sides of the equation by 2.

x = 4

Dividing isolates the variable

x

so we can find its numerical value.

3

Substitute

x = 4

into the second equation

2x + 3y = 17

and solve for

y

.

y = 3

Substituting the value of

x

leaves only the variable

y

, which can then be isolated and solved.

Anahtar Kavram

Solving systems of linear equations using elimination

Soru 3Soru →

Two linear equations are defined as follows:

3x + 2y = 12

x - 2y = 4

If the ordered pair $(x, y)$ satisfies both equations, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x

is 4.

Adding the two equations

3x + 2y = 12

and

x - 2y = 4

eliminates the

y

terms, resulting in

4x = 16

. Dividing both sides by 4 gives the value of

x

as 4.

Adım Adım Çözüm

1

Add the two equations together to eliminate the

y

variable.

4x = 16

Since the coefficients of

y

are opposites (

2

and

-2

), adding the equations eliminates

y

directly.

2

Divide both sides of the equation by 4.

x = 4

To isolate the variable

x

.

Anahtar Kavram

Solving systems of linear equations using the elimination method.

Alternatif Yöntem

We can solve the second equation for

x

to get

x = 2y + 4

. Substituting this expression into the first equation gives

3(2y + 4) + 2y = 12

, which simplifies to

6y + 12 + 2y = 12

, or

8y = 0

, meaning

y = 0

. Substituting

y = 0

back into

x = 2y + 4

yields

x = 4

.

Tahmini Süre:45s

Soru 4Soru →

Consider the system of equations below.

y = 3x - 4

x + y = 8

What is the value of $x$ ?

3

5

1

2

Cevabı ve açıklamayı göster

Cevap: $3$

Cevap

The value of

x

is

3

.

To solve the system, substitute the value of

y

from the first equation into the second equation:

x + (3x - 4) = 8

. Combining the

x

terms gives

4x - 4 = 8

. Adding

4

to both sides of the equation results in

4x = 12

. Dividing both sides by

4

isolates the variable, giving

x = 3

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation into the second equation.

x + (3x - 4) = 8

This eliminates the variable

y

to create a single equation in terms of

x

.

2

Combine the like terms on the left side of the equation.

4x - 4 = 8

Grouping the

x

variables simplifies the expression.

3

Add

4

to both sides of the equation to isolate the term with

x

.

4x = 12

Performing the inverse operation of subtraction isolates the variable term.

4

Divide both sides of the equation by

4

to solve for

x

.

x = 3

Performing the inverse operation of multiplication isolates the variable

x

.

Anahtar Kavram

Solving a system of linear equations using the substitution method.

Soru 5Soru →

If $(x, y)$ is the solution to the system of equations below, what is the value of $x$ ?

y = 2x + 1

3x + 2y = 16

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2.

Substituting

y = 2x + 1

into the second equation yields

3x + 2(2x + 1) = 16

. Expanding this expression gives

3x + 4x + 2 = 16

, which simplifies to

7x + 2 = 16

. Subtracting 2 from both sides gives

7x = 14

, and dividing by 7 yields

x = 2

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation into the second equation.

3x + 2(2x + 1) = 16

This reduces the system of two variables to a single equation in terms of

x

.

2

Expand and simplify the equation by distributing and combining like terms.

7x + 2 = 16

Simplifying the equation makes it easier to isolate the variable

x

.

3

Isolate the variable

x

by performing inverse operations.

x = 2

Subtracting 2 and then dividing by 7 isolates

x

to find its value.

Anahtar Kavram

Solving a system of linear equations using the substitution method.

Alternatif Yöntem

Alternatively, you can multiply the first equation by 2 to get

-4x + 2y = 2

and subtract it from

3x + 2y = 16

to eliminate

y

, which also yields

7x = 14

and thus

x = 2

.

Tahmini Süre:1m 0s

Soru 6Soru →

In the system of linear equations shown below, what value of $x$ satisfies both equations?

y = 2x - 5

3x - 2y = 12

-22

-2

2

-9

Cevabı ve açıklamayı göster

Cevap: $-2$

Cevap

-2

To find the value of

x

, substitute the expression for

y

from the first equation,

y = 2x - 5

, into the second equation,

3x - 2y = 12

. This yields

3x - 2(2x - 5) = 12

. Distributing the

-2

across the terms in the parentheses results in

3x - 4x + 10 = 12

. Simplifying the left side of the equation by combining like terms gives

-x + 10 = 12

. Subtracting

10

from both sides results in

-x = 2

. Finally, multiplying or dividing both sides by

-1

yields

x = -2

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation into the second equation

3x - 2(2x - 5) = 12

To eliminate

y

and obtain an equation containing only

x

.

2

Distribute

-2

to the terms inside the parentheses

3x - 4x + 10 = 12

To remove parentheses and prepare the equation for simplifying.

3

Combine like terms and solve for

x

-x + 10 = 12

, which gives

-x = 2

, and thus

x = -2

To isolate and find the value of

x

.

Anahtar Kavram

Solving systems of linear equations using substitution

Soru 7Soru →

In a science experiment, the temperature

y

, in degrees Celsius, and the time

x

, in minutes, are related by a system of equations. The relationships are given by:

y = 2x - 3

3x - y = 8

What is the value of

x

in the solution to this system?

1

5

7

11

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct answer is 5. Substituting the expression for

y

from the first equation into the second equation yields

3x - (2x - 3) = 8

. Distributing the negative sign results in

3x - 2x + 3 = 8

. Combining the variable terms gives

x + 3 = 8

. Subtracting 3 from both sides isolates

x

, giving

x = 5

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation,

y = 2x - 3

, into the second equation.

3x - (2x - 3) = 8

This eliminates the variable

y

so the equation can be solved directly for

x

.

2

Distribute the negative sign to both terms inside the parentheses.

3x - 2x + 3 = 8

Distributing the negative sign to both

2x

and

-3

simplifies the expression correctly.

3

Combine like terms on the left side of the equation.

x + 3 = 8

Subtracting

2x

from

3x

leaves a single

x

term.

4

Subtract 3 from both sides of the equation to isolate

x

.

x = 5

This isolates the variable

x

to find the final solution.

Anahtar Kavram

Solving systems of linear equations using substitution

Tahmini Süre:1m 0s

Soru 8Soru →

Consider the system of equations below.

x = 2y

3x - y = 10

If $(x, y)$ is the solution to the system of equations above, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x

is

4

.

The system of equations can be solved by substituting

x = 2y

from the first equation into the second equation, which gives

3(2y) - y = 10

. Simplifying this equation yields

6y - y = 10

, or

5y = 10

. Dividing by

5

gives

y = 2

. Substituting

y = 2

back into the first equation yields

x = 2(2) = 4

. Therefore, the value of

x

is

4

.

Adım Adım Çözüm

1

Substitute the expression for

x

from the first equation into the second equation.

3(2y) - y = 10

Since the first equation gives

x

in terms of

y

, substituting it into the second equation reduces the system to a single linear equation in one variable.

2

Solve the resulting equation for

y

.

y = 2

Simplifying

3(2y) - y = 10

yields

6y - y = 10

, which simplifies further to

5y = 10

. Dividing both sides by

5

gives

y = 2

.

3

Substitute the value of

y

back into the first equation to solve for

x

.

x = 4

Using

x = 2y

and substituting

y = 2

gives

x = 2(2) = 4

.

Anahtar Kavram

Solving systems of linear equations using the substitution method.

Soru 9Soru →

The variables $x$ and $y$ satisfy the system of equations below.

y = 10 - 2x

3x - y = 5

What is the value of $x$ ?

15

4

3

1

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

Substituting the expression for

y

from the first equation,

10 - 2x

, into the second equation yields

3x - (10 - 2x) = 5

. Distributing the negative sign results in

3x - 10 + 2x = 5

. Combining the like terms of

x

gives

5x - 10 = 5

. Adding 10 to both sides of the equation results in

5x = 15

. Finally, dividing both sides by 5 gives

x = 3

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation into the second equation.

3x - (10 - 2x) = 5

To eliminate the variable

y

and obtain a single linear equation in terms of

x

.

2

Distribute the negative sign through the parentheses and combine like terms.

3x - 10 + 2x = 5

, which simplifies to

5x - 10 = 5

To simplify the linear equation before isolating the variable.

3

Isolate the variable

x

by adding 10 to both sides and then dividing by 5.

5x = 15

, which gives

x = 3

To determine the value of

x

.

Anahtar Kavram

Solving systems of linear equations using substitution

Tahmini Süre:45s

Soru 10Soru →

In the system of equations below, what is the value of $x$ ?

3x - y = 7

x + y = 5

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is

3

.

Adding the two equations eliminates

y

, resulting in

4x = 12

. Dividing by

4

gives

x = 3

.

Adım Adım Çözüm

1

Add the two equations to eliminate

y

.

4x = 12

Adding the equations

(3x - y) + (x + y) = 7 + 5

eliminates

y

because

-y + y = 0

.

2

Solve for

x

.

x = 3

Divide both sides of the equation

4x = 12

by

4

.

Anahtar Kavram

Solving a system of linear equations by elimination.

Soru 11Soru →

In the $xy$ -plane, the graphs of the linear equations $y = 3x - 5$ and $y = -x + 7$ intersect at the point $(x, y)$ . What is the value of $x$ ?

1

3

4

6

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

To find the

x

-value of the intersection point of the two graphs, set the two equations equal to each other because both represent the same

y

-value at that point:

3x - 5 = -x + 7

. Adding

x

to both sides of the equation gives

4x - 5 = 7

. Adding

5

to both sides gives

4x = 12

. Dividing both sides by

4

yields

x = 3

.

Adım Adım Çözüm

1

Set the two expressions for

y

equal to each other to find the

x

-coordinate of the intersection point.

3x - 5 = -x + 7

Since both equations are solved for

y

, their right-hand sides must be equal at the point of intersection.

2

Add

x

to both sides of the equation to collect the variable terms on one side.

4x - 5 = 7

Moving the variable terms together allows us to isolate

x

.

3

Add

5

to both sides of the equation to isolate the term with

x

.

4x = 12

Grouping the constant terms on the opposite side prepares the equation for final division.

4

Divide both sides by

4

to find the value of

x

.

x = 3

This isolates

x

and gives the final solution.

Anahtar Kavram

Solving a system of linear equations by setting the equations equal to find the point of intersection in the coordinate plane.

Soru 12Soru →

A food truck sells tacos and burritos. On Tuesday, the food truck sold a total of 120 tacos and burritos. The number of tacos sold, $t$ , was 20 more than the number of burritos sold, $b$ . This situation is represented by the system of equations below:

t + b = 120

t = b + 20

How many burritos did the food truck sell on Tuesday?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The food truck sold 50 burritos on Tuesday.

To find the number of burritos sold,

b

, we substitute

t = b + 20

into the first equation

t + b = 120

, giving

(b + 20) + b = 120

. Combining like terms yields

2b + 20 = 120

. Subtracting 20 from both sides results in

2b = 100

. Finally, dividing both sides by 2 gives

b = 50

. Thus, the food truck sold 50 burritos on Tuesday.

Adım Adım Çözüm

1

Substitute the expression for

t

from the second equation into the first equation

(b + 20) + b = 120

To eliminate the variable

t

and set up an equation with only one variable,

b

.

2

Combine like terms on the left side of the equation

2b + 20 = 120

To simplify the equation for solving.

3

Subtract 20 from both sides of the equation

2b = 100

To isolate the variable term on one side of the equation.

4

Divide both sides of the equation by 2

b = 50

To solve for the variable

b

representing the number of burritos.

Anahtar Kavram

Solving systems of linear equations using the substitution method

Soru 13Soru →

A system of linear equations is shown below.

y = 2x - 7

3x - 2y = 11

What is the value of $x$ ?

8

-25

3

-3

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct value of

x

is 3.

The correct value of

x

is 3, which is found by substituting

2x - 7

for

y

in the second equation and solving for

x

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation into the second equation.

3x - 2(2x - 7) = 11

Since the first equation defines

y

in terms of

x

, substitution is a direct way to eliminate

y

and solve for

x

.

2

Distribute the

-2

to both terms inside the parentheses.

3x - 4x + 14 = 11

Applying the distributive property requires multiplying

-2

by both

2x

and

-7

, remembering that negative times negative is positive.

3

Combine the like terms on the left side of the equation.

-x + 14 = 11

Simplifying the expression by combining the

x

terms (

3x - 4x = -x

).

4

Isolate the variable term by subtracting 14 from both sides.

-x = -3

Using the subtraction property of equality to move the constant term to the right side.

5

Solve for

x

by multiplying or dividing both sides by

-1

.

x = 3

Finding the positive value of

x

.

Anahtar Kavram

Solving systems of linear equations using the substitution method.

Soru 14Soru →

If the equations $x + 2y = 12$ and $x + y = 7$ are true for the same values of $x$ and $y$ , what is the value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

y

is 5.

Subtracting the equation

x + y = 7

from

x + 2y = 12

gives

(x - x) + (2y - y) = 12 - 7

, which simplifies directly to

y = 5

.

Adım Adım Çözüm

1

Subtract the second equation from the first equation.

y = 5

Subtracting

x + y = 7

from

x + 2y = 12

eliminates the

x

terms, leaving a single linear equation in terms of

y

.

Anahtar Kavram

Solving systems of linear equations using elimination.

Soru 15Soru →

A system of two linear equations is given below.

y = x - 2

2x - 3y = 1

If $(x, y)$ is the solution to the system of equations, what is the value of $x$ ?

5

-7

-5

7

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

x

is 5.

To find the value of

x

, substitute the expression for

y

from the first equation into the second equation:

2x - 3(x - 2) = 1

. Distribute the

-3

to obtain

2x - 3x + 6 = 1

, which simplifies to

-x + 6 = 1

. Subtracting 6 from both sides gives

-x = -5

. Multiplying or dividing both sides by

-1

yields

x = 5

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation into the second equation.

2x - 3(x - 2) = 1

Substituting the expression reduces the system of two equations to one linear equation containing only the variable

x

.

2

Distribute the coefficient

-3

to both terms inside the parentheses.

2x - 3x + 6 = 1

Applying the distributive property removes the parentheses so that like terms can be combined.

3

Combine like terms and isolate

x

.

-x + 6 = 1 \implies -x = -5 \implies x = 5

Combining the

x

terms and subtracting 6 from both sides isolates

-x

, allowing you to find the final value of

x

.

Anahtar Kavram

Solving systems of linear equations using the substitution method

Alternatif Yöntem

Alternatively, you can solve this system using elimination. First, rewrite the first equation in standard form by subtracting

x

from both sides, which gives

-x + y = -2

. Next, multiply this equation by 3 to align coefficients:

-3x + 3y = -6

. Adding this result directly to the second equation,

2x - 3y = 1

, eliminates the

y

terms and leaves

-x = -5

, which simplifies to

x = 5

.

Tahmini Süre:45s

Soru 16Soru →

A system of equations is shown below.

y = -2(x - 5)

3x + y = 13

What is the value of $x$ ?

23

4

3

8

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is

3

.

Substituting the expression for

y

from the first equation into the second equation gives

3x - 2(x - 5) = 13

. Distributing the

-2

yields

3x - 2x + 10 = 13

. Combining the

x

terms gives

x + 10 = 13

. Subtracting 10 from both sides of the equation yields

x = 3

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation into the second equation.

3x - 2(x - 5) = 13

This substitution reduces the system to a single linear equation in terms of

x

.

2

Apply the distributive property to simplify the equation.

3x - 2x + 10 = 13

Multiplying

-2

by

x

gives

-2x

, and multiplying

-2

by

-5

gives

+10

.

3

Combine like terms and isolate

x

.

x + 10 = 13

, which simplifies to

x = 3

Subtracting 10 from both sides gives the value of

x

.

Anahtar Kavram

Solving a system of linear equations using substitution.

Tahmini Süre:1m 0s

Soru 17Soru →

In the system of equations below, $k$ is a constant.

kx - 3y = 4

(k-2)x - 5y = 8

If the system has no solution, what is the value of $k$ ?

-1

5

-3

3

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

-3

To find the value of

k

for which the system of linear equations has no solution, we determine when the two lines represented by the equations are parallel and distinct. Parallel lines have equal slopes. We can express each equation in slope-intercept form,

y = mx + b

. For the first equation,

kx - 3y = 4

simplifies to

y = \frac{k}{3}x - \frac{4}{3}

, giving a slope of

\frac{k}{3}

. For the second equation,

(k-2)x - 5y = 8

simplifies to

y = \frac{k-2}{5}x - \frac{8}{5}

, giving a slope of

\frac{k-2}{5}

. Equating the two slopes yields

\frac{k}{3} = \frac{k-2}{5}

. Cross-multiplying gives

5k = 3(k-2)

, which simplifies to

5k = 3k - 6

. Subtracting

3k

from both sides gives

2k = -6

, and dividing by

2

results in

k = -3

. Since the y-intercepts

-\frac{4}{3}

and

-\frac{8}{5}

are distinct, the lines are parallel and do not intersect, meaning there is no solution.

Adım Adım Çözüm

1

Express both equations in slope-intercept form (

y = mx + b

) to find their slopes.

For the first equation,

kx - 3y = 4 \implies 3y = kx - 4 \implies y = \frac{k}{3}x - \frac{4}{3}

(slope is

\frac{k}{3}

). For the second equation,

(k-2)x - 5y = 8 \implies 5y = (k-2)x - 8 \implies y = \frac{k-2}{5}x - \frac{8}{5}

(slope is

\frac{k-2}{5}

).

A system of two linear equations has no solution if the lines are parallel (slopes are equal) and their y-intercepts are different.

2

Set the slopes equal to each other.

\frac{k}{3} = \frac{k-2}{5}

Since the lines must be parallel, their slopes must be equivalent.

3

Solve for

k

by cross-multiplying.

5k = 3(k-2) \implies 5k = 3k - 6 \implies 2k = -6 \implies k = -3

.

Cross-multiplication isolates the variable

k

. Since the y-intercepts (

-\frac{4}{3}

and

-\frac{8}{5}

) are different, this value of

k

guarantees the lines are parallel and distinct.

Anahtar Kavram

Determining parameters for a system of linear equations with no solution

Alternatif Yöntem

Alternatively, align the coefficients of

y

to eliminate the

y

terms. Multiply the first equation by

5

and the second equation by

-3

:

5(kx - 3y) = 5(4) \implies 5kx - 15y = 20

-3((k-2)x - 5y) = -3(8) \implies -3(k-2)x + 15y = -24

Adding these two equations yields:

(5k - 3(k-2))x = -4

(2k + 6)x = -4

For the system to have no solution, the coefficient of

x

must be

0

while the constant on the right side is non-zero (which it is,

-4

). Setting the coefficient of

x

to

0

:

2k + 6 = 0 \implies k = -3

Tahmini Süre:2m 0s

Soru 18Soru →

\begin{aligned} 3x - 5y &= 8 \\ kx + 10y &= -3 \end{aligned}

In the system of equations above, $k$ is a constant. If the system has no solution, what is the value of $k$ ?

-2

-6

-1.5

6

Cevabı ve açıklamayı göster

Cevap: -6

Cevap

-6

The correct answer is

-6

. A system of two linear equations has no solution when the equations represent parallel lines, meaning they have the same slope but different

y

-intercepts. Writing the first equation in slope-intercept form gives

y = \frac{3}{5}x - \frac{8}{5}

, so its slope is

\frac{3}{5}

. Writing the second equation in slope-intercept form gives

y = -\frac{k}{10}x - \frac{3}{10}

, so its slope is

-\frac{k}{10}

. Since the

y

-intercepts are different (

-\frac{8}{5} \neq -\frac{3}{10}

), the system will have no solution when their slopes are equal:

\frac{3}{5} = -\frac{k}{10}

. Solving this equation for

k

yields

k = -6

.

Adım Adım Çözüm

1

Convert the first equation to slope-intercept form (

y = mx + b

) to find its slope.

y = \frac{3}{5}x - \frac{8}{5}

, so the slope is

\frac{3}{5}

and the

y

-intercept is

-\frac{8}{5}

.

Expressing the line in slope-intercept form directly reveals its slope and

y

-intercept.

2

Convert the second equation to slope-intercept form to express its slope in terms of

k

.

10y = -kx - 3 \Rightarrow y = -\frac{k}{10}x - \frac{3}{10}

, so the slope is

-\frac{k}{10}

and the

y

-intercept is

-\frac{3}{10}

.

This allows comparison of the slope and

y

-intercept of the second line with the first line.

3

Set the slopes equal to each other and solve for

k

.

\frac{3}{5} = -\frac{k}{10} \Rightarrow 30 = -5k \Rightarrow k = -6

.

A system of two linear equations has no solution if and only if the lines are parallel (equal slopes) and have different

y

-intercepts (which

-\frac{8}{5}

and

-\frac{3}{10}

are).

Anahtar Kavram

Determining the number of solutions to a system of linear equations based on slope and intercept relationships.

Soru 19Soru →

In the system of equations below, $a$ is a constant.

\begin{aligned} ax + 3y &= 5 \\ 2x - y &= 1 \end{aligned}

If the system has a solution $(x, y)$ that lies in Quadrant IV of the $xy$ -plane, which of the following could be the value of $a$ ?

12

8

-4

-8

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The correct answer is 12. For the solution

(x, y)

to lie in Quadrant IV, the

x

-coordinate must be positive and the

y

-coordinate must be negative. Isolating

y

in the second equation gives

y = 2x - 1

. Substituting this into the first equation yields

x = \frac{8}{a+6}

. For

x

to be positive, we must have

a > -6

. Substituting

x

back into the expression for

y

gives

y = \frac{10-a}{a+6}

. For

y

to be negative when

a > -6

, we must have

10 - a < 0

, which simplifies to

a > 10

. Among the choices, only 12 is greater than 10.

Adım Adım Çözüm

1

Identify the signs of coordinates in Quadrant IV.

x > 0

and

y < 0

In Quadrant IV of the coordinate plane, the horizontal coordinate is positive and the vertical coordinate is negative.

2

Express

y

in terms of

x

from the second equation.

y = 2x - 1

This facilitates substituting

y

into the first equation to solve for

x

.

3

Substitute

y = 2x - 1

into the first equation and solve for

x

.

x = \frac{8}{a + 6}

To find the coordinates of the intersection point in terms of the parameter

a

.

4

Substitute

x = \frac{8}{a + 6}

back into the expression for

y

.

y = \frac{10 - a}{a + 6}

To express the

y

-coordinate of the intersection point in terms of the parameter

a

.

5

Apply the conditions

x > 0

and

y < 0

to solve for

a

.

a > 10

For

x > 0

, we need

a + 6 > 0 \implies a > -6

. For

y < 0

given

a + 6 > 0

, we need

10 - a < 0 \implies a > 10

.

6

Evaluate the options to find a value of

a

that satisfies the condition.

12

Among the options (12, 8, -4, -8), only 12 is greater than 10.

Anahtar Kavram

Solving systems of linear equations with parameters and applying quadrant constraints

Soru 20Soru →

In the system of equations below, $a$ and $b$ are constants.

\begin{aligned} 3x - 4y &= 7 \\ ax - 12y &= b \end{aligned}

If the system has infinitely many solutions, what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

For a system of two linear equations in two variables to have infinitely many solutions, the two equations must represent the same line. Comparing the

y

-coefficients shows that

-12

is

3

times

-4

. Multiplying the first equation by

3

gives the equivalent equation

9x - 12y = 21

. For this to be identical to the second equation,

ax - 12y = b

, the corresponding coefficients must be equal, meaning

a = 9

and

b = 21

. Therefore, the value of

a + b

is

9 + 21 = 30

.

Adım Adım Çözüm

1

Identify the condition for a system of linear equations to have infinitely many solutions.

The two equations must represent the same line, making them equivalent equations.

Infinitely many solutions occur when the two equations in a system are scalar multiples of each other.

2

Align the coefficients of the

y

-terms in both equations.

Multiplying the first equation by

3

gives

9x - 12y = 21

.

The coefficient of

y

in the second equation is

-12

, which is

3

times the coefficient of

y

in the first equation (

-4

).

3

Equate the coefficients of the remaining terms to find the values of

a

and

b

.

a = 9

and

b = 21

.

For

9x - 12y = 21

and

ax - 12y = b

to be equivalent, their corresponding coefficients and constants must be equal.

4

Calculate the value of

a + b

.

a + b = 9 + 21 = 30

.

The problem asks for the value of the sum

a + b

.

Anahtar Kavram

Determining constants in a system of linear equations with infinitely many solutions

Tahmini Süre:1m 30s

Systems of Linear Equations

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