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Difficulty: HardLinear Equations and Graphing

In the standard (x,y)(x,y) coordinate plane, a line with a negative slope passes through the point (3,4)(3, 4). If the sum of the line's xx-intercept and yy-intercept is 1414, which of the following could be the slope of this line?

  1. A
    −34-\frac{3}{4}
  2. −43-\frac{4}{3}Answer
  3. C
    −23-\frac{2}{3}
  4. D
    −4-4
  5. E
    43\frac{4}{3}

Answer

−43-\frac{4}{3}
The correct answer is −43-\frac{4}{3}. By representing the line in point-slope form as y−4=m(x−3)y - 4 = m(x - 3), we determine that the yy-intercept is at (0,−3m+4)(0, -3m + 4) and the xx-intercept is at (3−4m,0)\left(3 - \frac{4}{m}, 0\right). Adding these intercepts together and setting the sum equal to 1414 gives the equation −3m+7−4m=14-3m + 7 - \frac{4}{m} = 14. Simplifying this equation leads to the quadratic expression 3m2+7m+4=03m^2 + 7m + 4 = 0, which factors as (3m+4)(m+1)=0(3m + 4)(m + 1) = 0. This yields two possible negative slopes: −43-\frac{4}{3} and −1-1. Among the choices, −43-\frac{4}{3} is the only matching option.

Step-by-Step Solution

1
Write the general equation of a line passing through (3,4)(3, 4) with slope mm.
y−4=m(x−3)y - 4 = m(x - 3)
Using the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) is a direct way to parameterize the line in terms of its slope.
2
Find the yy-intercept of the line by setting x=0x = 0.
y=−3m+4y = -3m + 4
The yy-intercept is the point where the line crosses the yy-axis, which occurs when x=0x = 0.
3
Find the xx-intercept of the line by setting y=0y = 0 and solving for xx.
x=3−4mx = 3 - \frac{4}{m}
The xx-intercept is the point where the line crosses the xx-axis, which occurs when y=0y = 0.
4
Set the sum of the xx-intercept and yy-intercept equal to 1414.
(−3m+4)+(3−4m)=14(-3m + 4) + \left(3 - \frac{4}{m}\right) = 14
This translates the given condition that the sum of the intercepts is 1414.
5
Simplify the equation and clear the fraction by multiplying by −m-m.
3m2+7m+4=03m^2 + 7m + 4 = 0
Grouping like terms yields −3m−4m−7=0-3m - \frac{4}{m} - 7 = 0. Multiplying by −m-m transforms it into a standard quadratic equation.
6
Factor the quadratic equation to solve for mm.
(3m+4)(m+1)=0  ⟹  m=−43(3m + 4)(m + 1) = 0 \implies m = -\frac{4}{3} or m=−1m = -1
Factoring helps find the values of mm that satisfy the equation. Both values represent lines with negative slopes.

Key Concept

Using the point-slope form of a linear equation to find intercepts and solving the resulting quadratic equation to determine the slope.

Alternative Method

An alternative approach is using the intercept form of a line, xa+yb=1\frac{x}{a} + \frac{y}{b} = 1, where aa and bb are the xx- and yy-intercepts. Since the point (3,4)(3, 4) lies on the line, we have 3a+4b=1\frac{3}{a} + \frac{4}{b} = 1. We are also given a+b=14  ⟹  b=14−aa + b = 14 \implies b = 14 - a. Substituting bb into the equation gives 3a+414−a=1\frac{3}{a} + \frac{4}{14-a} = 1. Solving this equation by finding a common denominator results in 3(14−a)+4a=a(14−a)  ⟹  42+a=14a−a2  ⟹  a2−13a+42=03(14-a) + 4a = a(14-a) \implies 42 + a = 14a - a^2 \implies a^2 - 13a + 42 = 0. Factoring gives (a−6)(a−7)=0(a-6)(a-7) = 0, so a=6a = 6 or a=7a = 7. If a=6a = 6, then b=8b = 8, and the slope m=−ba=−86=−43m = -\frac{b}{a} = -\frac{8}{6} = -\frac{4}{3}. If a=7a = 7, then b=7b = 7, and the slope m=−77=−1m = -\frac{7}{7} = -1. This confirms the possible slopes.
Estimated Time:2m 30s
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Topics

Linear Equations and Graphing

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