In the xy-plane, the graph of the linear equation ax + by = c, where a, b, and c are const… — SAT

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has the property that the value of $y$ decreases by $3$ for every increase of $5$ in the value of $x$ . If the graph of this equation passes through the point $(4, -2)$ and $a = 6$ , what is the value of $c$ ?

Answer: 4

Answer

The correct answer is

4

The correct answer is

4

. The slope of the line described is

m = \frac{\Delta y}{\Delta x} = \frac{-3}{5} = -0.6

. Rearranging

ax + by = c

into slope-intercept form

y = -\frac{a}{b}x + \frac{c}{b}

shows that the slope is

-\frac{a}{b}

. Setting these equal gives

-\frac{a}{b} = -\frac{3}{5} \implies \frac{a}{b} = \frac{3}{5}

. Since

a = 6

, we find

b = 10

. Substituting

a = 6

b = 10

, and the coordinates of the point

(4, -2)

into the original equation yields

6(4) + 10(-2) = 4

Step-by-Step Solution

Determine the slope of the line from the described relationship between

x

and

y

The slope

m

of the line is

-\frac{3}{5}

The slope represents the change in

y

divided by the change in

x

. Since

y

decreases by

3

(

\Delta y = -3

) for every increase of

5

x

(

\Delta x = 5

), the slope is

m = \frac{\Delta y}{\Delta x} = -\frac{3}{5}

Express the slope of the line in terms of the coefficients from the standard form equation

ax + by = c

The slope of the line is

-\frac{a}{b}

Rewriting the equation

ax + by = c

in slope-intercept form gives

by = -ax + c \implies y = -\frac{a}{b}x + \frac{c}{b}

. The coefficient of

x

is the slope, so

m = -\frac{a}{b}

Equate the two expressions for the slope and solve for

b

using the given value of

a = 6

b = 10

Setting

-\frac{a}{b} = -\frac{3}{5}

gives

\frac{a}{b} = \frac{3}{5}

. Substituting

a = 6

yields

\frac{6}{b} = \frac{3}{5}

, which simplifies to

3b = 30

, so

b = 10

Substitute the point

(4, -2)

and the values of

a

and

b

into the equation

ax + by = c

to solve for

c

c = 4

Substituting

a = 6

b = 10

x = 4

, and

y = -2

into

ax + by = c

gives

6(4) + 10(-2) = c \implies 24 - 20 = c \implies c = 4

Key Concept

Linear Equations in Two Variables

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