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Difficulty: MediumSlope of a Line

In the standard (x,y)(x, y) coordinate plane, a line segment has endpoints A(−2,3)A(-2, 3) and B(4,y)B(4, y). If the slope of the line passing through AA and BB is −23-\frac{2}{3}, what is the value of yy?

  1. A
    -7
  2. B
    -6
  3. -1Answer
  4. D
    53\frac{5}{3}
  5. E
    7

Answer

The value of yy is −1-1.
The correct option is the one showing −1-1. By applying the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} to the points A(−2,3)A(-2, 3) and B(4,y)B(4, y) with slope −23-\frac{2}{3}, we set up the equation y−34−(−2)=−23\frac{y - 3}{4 - (-2)} = -\frac{2}{3}. Simplifying the denominator yields y−36=−23\frac{y - 3}{6} = -\frac{2}{3}. Multiplying both sides by 66 gives y−3=−4y - 3 = -4, and solving for yy yields y=−1y = -1.

Step-by-Step Solution

1
Write down the slope formula and substitute the given values.
y−34−(−2)=−23\frac{y - 3}{4 - (-2)} = -\frac{2}{3}
The slope mm of a line passing through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is defined as m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
2
Simplify the denominator on the left side of the equation.
y−36=−23\frac{y - 3}{6} = -\frac{2}{3}
Subtracting a negative number is equivalent to addition: 4−(−2)=4+2=64 - (-2) = 4 + 2 = 6.
3
Multiply both sides by 6 to isolate the numerator.
y−3=−4y - 3 = -4
Multiplying −23-\frac{2}{3} by 66 yields −4-4.
4
Solve for yy by adding 3 to both sides of the equation.
y=−1y = -1
Isolating yy gives y=−4+3=−1y = -4 + 3 = -1.

Key Concept

Using the slope formula to find a missing coordinate
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Topics

Slope of a Line

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