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Difficulty: MediumLinear Equations and Graphing

In the standard (x,y)(x, y) coordinate plane, a line passes through the point (3,2)(3, 2) and has a yy-intercept of −4-4. If the point (k,5k+2)(k, 5k + 2) also lies on this line, what is the value of kk?

Answer: -2

Answer

The value of kk is −2-2.
The line has a slope of 22 and a y-intercept of −4-4, giving the equation y=2x−4y = 2x - 4. Substituting the coordinates of (k,5k+2)(k, 5k + 2) results in 5k+2=2k−45k + 2 = 2k - 4, which simplifies to 3k=−63k = -6, yielding k=−2k = -2.

Step-by-Step Solution

1
Identify the coordinates of the y-intercept.
The y-intercept of −4-4 corresponds to the point (0,−4)(0, -4).
The y-intercept is the point where the line crosses the y-axis, meaning the x-coordinate is 0.
2
Calculate the slope of the line.
The slope mm is 22.
Using the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} with the points (3,2)(3, 2) and (0,−4)(0, -4) gives m=2−(−4)3−0=63=2m = \frac{2 - (-4)}{3 - 0} = \frac{6}{3} = 2.
3
Write the equation of the line.
The equation of the line is y=2x−4y = 2x - 4.
Using the slope-intercept form y=mx+by = mx + b, where the slope m=2m = 2 and the y-intercept b=−4b = -4.
4
Substitute the point (k,5k+2)(k, 5k + 2) into the line's equation.
The equation becomes 5k+2=2k−45k + 2 = 2k - 4.
Since the point lies on the line, its coordinates must satisfy the line's equation.
5
Solve the linear equation for kk.
k=−2k = -2.
Subtracting 2k2k from both sides gives 3k+2=−43k + 2 = -4. Subtracting 22 from both sides gives 3k=−63k = -6. Dividing by 33 gives k=−2k = -2.

Key Concept

Finding the equation of a line from a point and an intercept, and solving for parameters of points on that line.
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Topics

Linear Equations and Graphing

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