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 a slope of $\frac{2}{3}$ and passes through the point $(6, 5)$ . If $a + b = 5$ , what is the value of $c$ ?

Answer: 15

Answer

Rearranging the linear equation

ax + by = c

into slope-intercept form

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

shows the slope is

-\frac{a}{b}

. Setting

-\frac{a}{b} = \frac{2}{3}

gives

2b = -3a

, or

b = -1.5a

. Substituting this into

a + b = 5

yields

a - 1.5a = 5

, which simplifies to

-0.5a = 5

, so

a = -10

. This means

b = -1.5(-10) = 15

. Substituting the values of

a = -10

and

b = 15

along with the point

(6, 5)

into the equation

ax + by = c

yields

(-10)(6) + (15)(5) = 15

, so the value of

c

15

Step-by-Step Solution

Rewrite the standard form equation

ax + by = c

in slope-intercept form.

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

This allows the identification of the slope of the line in terms of the coefficients

a

and

b

Set the slope expression equal to the given slope of

\frac{2}{3}

and solve for

b

in terms of

a

b = -1.5a

The slope of the line is

-\frac{a}{b}

, so

-\frac{a}{b} = \frac{2}{3} \implies 2b = -3a \implies b = -1.5a

Substitute

b = -1.5a

into the given equation

a + b = 5

and solve for

a

, then find

b

a = -10

and

b = 15

Substituting gives

a - 1.5a = 5 \implies -0.5a = 5 \implies a = -10

. Substituting

a = -10

back into the relationship gives

b = -1.5(-10) = 15

Substitute

a = -10

b = 15

, and the point

(6, 5)

into the equation

ax + by = c

and solve for

c

c = 15

(-10)(6) + (15)(5) = c \implies -60 + 75 = 15 \implies c = 15

Key Concept

Linear Equations in Two Variables

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