In the xy-plane, a line is represented by the equation ay - bx = 24, where a and b are con… — SAT

In the $xy$ -plane, a line is represented by the equation $ay - bx = 24$ , where $a$ and $b$ are constants. If the line has a $y$ -intercept of $(0, -3)$ and passes through the point $(4, 5)$ , what is the slope of the line?

Answer: 2

Answer

The slope of the line is 2.

Substituting the y-intercept

(0, -3)

into the equation

ay - bx = 24

yields

-3a = 24

, which gives

a = -8

. Substituting the point

(4, 5)

and

a = -8

into the equation yields

-8(5) - 4b = 24

, which simplifies to

-40 - 4b = 24

, giving

b = -16

. Re-assembling the equation gives

-8y + 16x = 24

. Solving for

y

in terms of

x

yields

y = 2x - 3

, where the coefficient of

x

is the slope, 2.

Step-by-Step Solution

Substitute the y-intercept

(0, -3)

into the given equation

ay - bx = 24

to find the value of

a

a = -8

Since the y-intercept lies on the line, its coordinates satisfy the line's equation.

Substitute the point

(4, 5)

and the value of

a = -8

into the equation to find the value of

b

b = -16

Since the point

(4, 5)

lies on the line, its coordinates must satisfy the equation.

Write the resulting equation

-8y + 16x = 24

in slope-intercept form (

y = mx + c

) to identify the slope.

y = 2x - 3

, which gives a slope of

2

The slope of a line in the form

y = mx + c

is represented by the coefficient

m

x

Key Concept

Finding the slope of a line from a given equation by determining its constant coefficients using known points.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic