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Difficulty: HardLinear Equations and Graphing

In the standard (x,y)(x, y) coordinate plane, a line with a non-zero slope mm passes through the point (8,2)(8, 2). If the product of the line's xx-intercept and its yy-intercept is 6464, what is the value of mm?

Answer: -0.25

Answer

The value of the slope mm is −0.25-0.25 (or −14-\frac{1}{4}).
By writing the line as y=mx+by = mx + b, the given point (8,2)(8, 2) establishes that b=2−8mb = 2 - 8m. Since the xx-intercept is −bm-\frac{b}{m}, the product of the intercepts is −b2m-\frac{b^2}{m}. Setting this product to 6464 gives −(2−8m)2m=64-\frac{(2-8m)^2}{m} = 64, which simplifies to the quadratic equation 16m2+8m+1=016m^2 + 8m + 1 = 0. Factoring this perfect square trinomial gives (4m+1)2=0(4m + 1)^2 = 0, which yields the unique solution m=−0.25m = -0.25.

Step-by-Step Solution

1
Express the yy-intercept in terms of mm.
b=2−8mb = 2 - 8m
The line equation is y=mx+by = mx + b and it passes through (8,2)(8, 2), so 2=8(m)+b2 = 8(m) + b.
2
Express the xx-intercept in terms of mm.
x=−bmx = -\frac{b}{m}
The xx-intercept is the value of xx when y=0y = 0 in the line equation y=mx+by = mx + b.
3
Write the product of the intercepts and set it equal to 6464.
b2=−64mb^2 = -64m
The product of the intercepts is b⋅(−bm)=−b2m=64b \cdot \left(-\frac{b}{m}\right) = -\frac{b^2}{m} = 64.
4
Substitute b=2−8mb = 2 - 8m into the product equation and simplify to a quadratic equation.
64m2+32m+4=064m^2 + 32m + 4 = 0
Substituting bb gives (2−8m)2=−64m(2 - 8m)^2 = -64m. Expanding and combining like terms yields the quadratic equation.
5
Solve the quadratic equation for mm.
m=−0.25m = -0.25
Dividing by 44 gives 16m2+8m+1=(4m+1)2=016m^2 + 8m + 1 = (4m + 1)^2 = 0, which has the single real solution m=−0.25m = -0.25.

Key Concept

Using linear equation forms, point-slope relationships, and intercept properties to solve coordinate geometry problems.
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Topics

Linear Equations and Graphing

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