The table below shows several values of x and the corresponding values for the linear functions f and g. | x | f(x) | g(x) | | :--- | :--- | :--- | | 1 | 7 | 17 | | 2 | 9 | 14 | | 4 | 13 | 8 | If (x, y) is the solution to the system of equations formed by y = f(x) and y = g(x), what is the value of

The table below shows several values of x and the corresponding values for the linear func… — SAT

The table below shows several values of $x$ and the corresponding values for the linear functions $f$ and $g$ .

$x$	$f(x)$	$g(x)$
1	7	17
2	9	14
4	13	8

If $(x, y)$ is the solution to the system of equations formed by $y = f(x)$ and $y = g(x)$ , what is the value of $2x + y$ ?

A
5
17Cevap
C
15
D
25

Cevap

The correct value of the expression is 17.

The correct answer is the value obtained by finding the linear equations for both functions, solving the resulting system of equations, and substituting the coordinates into the expression. The equation for the first function is

y = 2x + 5

and the equation for the second function is

y = -3x + 20

. Equating the two expressions yields

2x + 5 = -3x + 20

, which simplifies to

5x = 15

x = 3

. Substituting

x = 3

back into either equation yields

y = 11

. Evaluating the expression

2x + y

with these values gives

2(3) + 11 = 17

Adım Adım Çözüm

Determine the linear equation for

f(x)

using the points in the table.

The slope of

f(x)

\frac{9 - 7}{2 - 1} = 2

. Using the point-slope form with

(1, 7)

, the equation is

y - 7 = 2(x - 1)

, which simplifies to

y = 2x + 5

Finding the equation of the first line is necessary to set up the system of linear equations.

Determine the linear equation for

g(x)

using the points in the table.

The slope of

g(x)

\frac{14 - 17}{2 - 1} = -3

. Using the point-slope form with

(1, 17)

, the equation is

y - 17 = -3(x - 1)

, which simplifies to

y = -3x + 20

Finding the equation of the second line completes the system of equations.

Find the intersection point of

y = f(x)

and

y = g(x)

by setting the two equations equal to each other.

2x + 5 = -3x + 20 \Rightarrow 5x = 15 \Rightarrow x = 3

. Substituting

x = 3

into

y = 2x + 5

gives

y = 2(3) + 5 = 11

Solving the system of equations yields the values of

x

and

y

at the intersection point.

Calculate the value of

2x + y

using the solution

(3, 11)

2(3) + 11 = 6 + 11 = 17

This evaluates the specific linear combination requested in the question.

Anahtar Kavram

Solving systems of linear equations by translating tabular data into linear equations and finding their point of intersection.

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Cevap

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