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Difficulty: MediumLinear Equations in Two Variables

In the xyxy-plane, the graph of the linear equation y−4=m(x−6)y - 4 = m(x - 6), where mm is a constant, has a yy-intercept of (0,7)(0, 7). What is the xx-coordinate of the xx-intercept of the line?

  1. A
    -4
  2. B
    -2
  3. 14Answer
  4. D
    8

Answer

The correct answer is 14. This is the x-coordinate of the point where the line crosses the x-axis, which is (14, 0).
To find the value of mm, substitute the y-intercept (0,7)(0, 7) into the equation: 7−4=m(0−6)7 - 4 = m(0 - 6), which simplifies to 3=−6m3 = -6m, so m=−12m = -\frac{1}{2}. The equation of the line is y−4=−12(x−6)y - 4 = -\frac{1}{2}(x - 6). The x-intercept occurs when y=0y = 0. Substituting y=0y = 0 gives −4=−12(x−6)-4 = -\frac{1}{2}(x - 6), which simplifies to 8=x−68 = x - 6, so x=14x = 14.

Step-by-Step Solution

1
Substitute the y-intercept (0,7)(0, 7) into the given equation to solve for the constant mm.
7−4=m(0−6)  ⟹  3=−6m  ⟹  m=−127 - 4 = m(0 - 6) \implies 3 = -6m \implies m = -\frac{1}{2}
The y-intercept point (0,7)(0, 7) must satisfy the equation of the line, allowing us to find the slope mm.
2
Write the linear equation using the calculated value of mm.
y−4=−12(x−6)y - 4 = -\frac{1}{2}(x - 6)
Knowing the slope allows us to write the complete linear equation in point-slope form.
3
Find the x-intercept of the line by setting y=0y = 0 and solving for xx.
0−4=−12(x−6)  ⟹  −4=−12(x−6)  ⟹  8=x−6  ⟹  x=140 - 4 = -\frac{1}{2}(x - 6) \implies -4 = -\frac{1}{2}(x - 6) \implies 8 = x - 6 \implies x = 14
The x-intercept is the point on the line where the y-coordinate is equal to 0.

Key Concept

Finding the intercepts and slope of a linear equation in two variables.

Alternative Method

Alternatively, write the equation in slope-intercept form y=mx+by = mx + b. Since the y-intercept is (0,7)(0, 7), we have b=7b = 7, so the equation is y=mx+7y = mx + 7. Since the line passes through the point (6,4)(6, 4) (determined by the point-slope form y−4=m(x−6)y - 4 = m(x - 6)), substitute this point into the equation: 4=m(6)+7  ⟹  −3=6m  ⟹  m=−124 = m(6) + 7 \implies -3 = 6m \implies m = -\frac{1}{2}. Thus, the equation is y=−12x+7y = -\frac{1}{2}x + 7. Setting y=0y = 0 for the x-intercept gives 0=−12x+7  ⟹  12x=7  ⟹  x=140 = -\frac{1}{2}x + 7 \implies \frac{1}{2}x = 7 \implies x = 14.
Estimated Time:1m 30s
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Topics

Linear Equations in Two Variables

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