In the standard (x, y) coordinate plane, a line passes through the point (3, 1) and has a … — ACT

In the standard $(x, y)$ coordinate plane, a line passes through the point $(3, 1)$ and has a slope of $-\frac{2}{3}$ . If the line intersects the $x$ -axis at the point $(p, 0)$ , what is the value of $p$ ?

Answer: 4.5

Answer

The value of

p

4.5

The line equation is

y - 1 = -\frac{2}{3}(x - 3)

. By substituting

y = 0

for the

x

-intercept, the equation becomes

-1 = -\frac{2}{3}(p - 3)

. Multiplying both sides by

-3

gives

3 = 2(p - 3)

, which simplifies to

2p = 9

, resulting in

p = 4.5

Step-by-Step Solution

Write the point-slope equation of the line.

y - 1 = -\frac{2}{3}(x - 3)

We use the point-slope form

y - y_1 = m(x - x_1)

with the point

(3, 1)

and slope

-\frac{2}{3}

Substitute the point

(p, 0)

into the line's equation.

-1 = -\frac{2}{3}(p - 3)

The

x

-intercept

(p, 0)

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

Solve for the variable

p

p = 4.5

Multiply by

-3

to get

3 = 2(p - 3)

, add

6

to both sides to get

2p = 9

, and divide by

2

Key Concept

Linear equations and graphing, specifically point-slope form and x-intercept calculation.

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