In the xy-plane, a line passes through the point (5, -1) and has a slope of \frac{2}{5}. I… — SAT

In the $xy$ -plane, a line passes through the point $(5, -1)$ and has a slope of $\frac{2}{5}$ . If the line also passes through the point $(15, p)$ , what is the value of $p$ ?

Answer: 3

Answer

The value of

p

3

To find the value of

p

, the equation of the line can be established using the point-slope form:

y - y_1 = m(x - x_1)

. Substituting the given slope

m = \frac{2}{5}

and the point

(5, -1)

gives

y - (-1) = \frac{2}{5}(x - 5)

. Simplifying this yields

y + 1 = \frac{2}{5}x - 2

, which reduces to

y = \frac{2}{5}x - 3

. Substituting the point

(15, p)

into this equation gives

p = \frac{2}{5}(15) - 3 = 6 - 3 = 3

Step-by-Step Solution

Determine the equation of the line using point-slope form.

y = \frac{2}{5}x - 3

The equation of a line with slope

m

passing through a point

(x_1, y_1)

y - y_1 = m(x - x_1)

. Substituting

m = \frac{2}{5}

and the point

(5, -1)

gives

y - (-1) = \frac{2}{5}(x - 5)

. Simplifying this equation results in

y + 1 = \frac{2}{5}x - 2

, which becomes

y = \frac{2}{5}x - 3

Substitute the point

(15, p)

into the linear equation.

p = 3

Since the line passes through the point

(15, p)

, the coordinates must satisfy the equation of the line. Substituting

x = 15

and

y = p

into

y = \frac{2}{5}x - 3

gives

p = \frac{2}{5}(15) - 3

, which simplifies to

p = 6 - 3 = 3

Key Concept

Using the slope and a point on a line to find another coordinate along the same line.

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