In the standard (x, y) coordinate plane, a line is defined by the equation 2x + 3y = 22. A… — ACT

In the standard $(x, y)$ coordinate plane, a line is defined by the equation $2x + 3y = 22$ . A second line passes through the origin $(0, 0)$ and intersects the first line at a point where $x = 5$ . What is the slope of this second line?

Answer: 0.8

Answer

The slope of the second line is

0.8

(or

\frac{4}{5}

The intersection point has an

x

-coordinate of

5

. Substituting this into

2x + 3y = 22

gives

2(5) + 3y = 22

, which simplifies to

3y = 12

, or

y = 4

. Thus, the intersection point is

(5, 4)

. The second line passes through

(0, 0)

and

(5, 4)

. Using the slope formula, the slope is

\frac{4 - 0}{5 - 0} = 0.8

Step-by-Step Solution

Substitute

x = 5

into the equation of the first line to find the

y

-coordinate of the intersection point.

The intersection point is

(5, 4)

Since the two lines intersect at

x = 5

, the intersection point must satisfy the equation of the first line.

Use the slope formula to find the slope of the line connecting

(0, 0)

and

(5, 4)

The slope is

0.8

The second line passes through the origin

(0, 0)

and the intersection point

(5, 4)

, so its slope is the ratio of the change in

y

to the change in

x

Key Concept

Finding the slope of a line given two points on the coordinate plane, where one point is determined by the intersection of two linear paths.

Alternative Method

Since the second line passes through the origin

(0, 0)

, its equation is of the form

y = mx

, where

m

is the slope. At the intersection point

(5, y)

, we have

y = 5m

. We can substitute this directly into the first line's equation:

2x + 3y = 22 \Rightarrow 2(5) + 3(5m) = 22 \Rightarrow 10 + 15m = 22 \Rightarrow 15m = 12 \Rightarrow m = \frac{12}{15} = 0.8

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Topics

More questions from this topic