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Difficulty: HardLinear Functions and Graphs

A line in the xyxy-plane has a slope of −3-3. The line passes through the point (2,k)(2, k) and has an xx-intercept of (r,0)(r, 0), where kk and rr are constants. If r+k=10r + k = 10, what is the yy-intercept of the line?

  1. A
    (0,4)(0, 4)
  2. B
    (0,6)(0, 6)
  3. C
    (0,10)(0, 10)
  4. (0,12)(0, 12)Answer

Answer

The y-intercept of the line is (0,12)(0, 12).
The correct answer is (0,12)(0, 12). By using the formula for slope m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} with the points (2,k)(2, k) and (r,0)(r, 0), we obtain −3=k−02−r-3 = \frac{k - 0}{2 - r}. Multiplying both sides by 2−r2 - r gives k=−3(2−r)k = -3(2 - r), which simplifies to k=−6+3rk = -6 + 3r, or 3r−k=63r - k = 6. We can solve this alongside the given equation r+k=10r + k = 10 by adding the two equations: (3r−k)+(r+k)=6+10  ⟹  4r=16(3r - k) + (r + k) = 6 + 10 \implies 4r = 16, which yields r=4r = 4. Substituting r=4r = 4 back into the sum equation gives k=6k = 6. Now, using the x-intercept point (4,0)(4, 0) and the slope of −3-3, the equation of the line in point-slope form is y−0=−3(x−4)y - 0 = -3(x - 4), which simplifies to y=−3x+12y = -3x + 12. The y-intercept of this line is found by setting x=0x = 0, yielding (0,12)(0, 12).

Step-by-Step Solution

1
Express the slope of the line using the coordinates of the two given points, (2,k)(2, k) and the x-intercept (r,0)(r, 0).
Using the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}, we set up the equation: −3=k−02−r-3 = \frac{k - 0}{2 - r}.
This connects the geometric concept of slope to the algebraic coordinates of the points on the line.
2
Simplify the slope equation to relate kk and rr.
Multiplying both sides by 2−r2 - r gives k=−3(2−r)  ⟹  k=−6+3rk = -3(2 - r) \implies k = -6 + 3r, which can be rewritten as 3r−k=63r - k = 6.
Simplifying the equation makes it easier to solve as part of a system of linear equations.
3
Solve the system of equations consisting of 3r−k=63r - k = 6 and the given equation r+k=10r + k = 10.
Adding the two equations together: (3r−k)+(r+k)=6+10  ⟹  4r=16  ⟹  r=4(3r - k) + (r + k) = 6 + 10 \implies 4r = 16 \implies r = 4. Substituting r=4r = 4 into r+k=10r + k = 10 gives 4+k=10  ⟹  k=64 + k = 10 \implies k = 6.
Solving the system of equations determines the numerical values of the constants rr and kk.
4
Find the equation of the line using the slope m=−3m = -3 and the x-intercept (4,0)(4, 0), then determine the y-intercept.
Using point-slope form: y−0=−3(x−4)  ⟹  y=−3x+12y - 0 = -3(x - 4) \implies y = -3x + 12. The y-intercept occurs when x=0x = 0, giving y=12y = 12, which corresponds to the point (0,12)(0, 12).
Writing the full linear equation allows us to find the y-intercept by evaluating the function at x=0x = 0.

Key Concept

Writing linear equations from given points and slope, and solving a system of linear equations to identify intercepts.
Estimated Time:2m 0s
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Topics

Linear Functions and Graphs

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