In the xy-plane, the system of equations below has infinitely many solutions: \begin{align… — SAT

In the $xy$ -plane, the system of equations below has infinitely many solutions:

\begin{aligned} \frac{1}{2}(ax - by) &= 7 \\ 3x - 2y &= c \end{aligned}

where $a$ , $b$ , and $c$ are constants. If the line representing the first equation passes through the point $(4, 1)$ , what is the value of $c$ ?

A
5
B
8
10Cevap
D
20

Cevap

The correct answer is 10. For a system of linear equations in two variables to have infinitely many solutions, the two equations must describe the exact same line, meaning their coefficients and constant terms are proportional. Multiplying the first equation by

2

yields

ax - by = 14

. Comparing this with

3x - 2y = c

shows that

\frac{a}{3} = \frac{b}{2} = \frac{14}{c}

, which simplifies to

b = \frac{2}{3}a

and

c = \frac{42}{a}

. Since the first line passes through

(4, 1)

, we substitute

x = 4

and

y = 1

to get

4a - b = 14

. Substituting

b = \frac{2}{3}a

gives

4a - \frac{2}{3}a = 14

, which simplifies to

\frac{10}{3}a = 14

, or

a = 4.2

. Finally, solving for

c

gives

c = \frac{42}{4.2} = 10

Adım Adım Çözüm

Clear the fraction in the first equation by multiplying both sides by

2

ax - by = 14

This puts the first equation into standard form, making it easier to compare with the second equation.

Set up the proportionality of the coefficients for the two equations to represent the same line.

\frac{a}{3} = \frac{-b}{-2} = \frac{14}{c} \implies b = \frac{2}{3}a

and

c = \frac{42}{a}

For a system of two linear equations to have infinitely many solutions, the equations must be equivalent, meaning their corresponding coefficients and constants must be proportional.

Substitute the given point

(4, 1)

into the first equation.

a(4) - b(1) = 14 \implies 4a - b = 14

Since the line passes through the point

(4, 1)

, the coordinates must satisfy the equation of the line.

Substitute

b = \frac{2}{3}a

into

4a - b = 14

and solve for

a

4a - \frac{2}{3}a = 14 \implies \frac{10}{3}a = 14 \implies a = 4.2

This reduces the equation to a single variable, allowing us to find the value of the parameter

a

Substitute

a = 4.2

into the expression for

c

to find its value.

c = \frac{42}{4.2} = 10

This uses the coefficient proportionality relation from step 2 to determine the constant term of the second equation.

Anahtar Kavram

Systems of linear equations with infinitely many solutions require the equations to represent the same line, meaning their coefficients and constant terms are proportional.

Soru

Cevap

Adım Adım Çözüm

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