In the xy-plane, the graph of the linear equation 3x + 5y = c, where c is a constant, pass… — SAT

In the $xy$ -plane, the graph of the linear equation $3x + 5y = c$ , where $c$ is a constant, passes through the point $(4, 3)$ . What is the $x$ -coordinate of the point on this graph where the $y$ -coordinate is $-3$ ?

Answer: 14

Answer

The correct answer is 14.

To find the correct answer, first substitute the given point

(4, 3)

into the equation

3x + 5y = c

to determine the value of the constant

c

3(4) + 5(3) = 12 + 15 = 27

. Thus, the equation is

3x + 5y = 27

. Next, substitute

-3

for

y

in this equation to find the corresponding

x

-coordinate:

3x + 5(-3) = 27 \implies 3x - 15 = 27

. Adding

15

to both sides gives

3x = 42

, and dividing by

3

yields

x = 14

Step-by-Step Solution

Substitute the point

(4, 3)

into the equation

3x + 5y = c

to solve for

c

c = 27

Since the point lies on the graph of the equation, its coordinates must satisfy the equation.

Substitute

y = -3

and

c = 27

into the equation

3x + 5y = c

3x - 15 = 27

We want to find the

x

-coordinate of the point on the line when the

y

-coordinate is

-3

Solve the equation

3x - 15 = 27

for

x

x = 14

Isolating

x

gives the

x

-coordinate of the point.

Key Concept

Using a known point on a line to find a constant coefficient or constant term, and using the resulting equation to find other coordinates.

Alternative Method

Alternatively, you can write the equation in slope-intercept form. Solving

3x + 5y = c

for

y

gives

y = -\frac{3}{5}x + \frac{c}{5}

. The slope of the line is

-\frac{3}{5}

. Since the slope is constant, the change in

y

divided by the change in

x

between

(4, 3)

and

(x, -3)

is equal to the slope:

\frac{-3 - 3}{x - 4} = -\frac{3}{5} \implies \frac{-6}{x - 4} = -\frac{3}{5}

. Cross-multiplying gives

-30 = -3(x - 4) \implies 10 = x - 4 \implies x = 14

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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