Algebra Session 6: Solving Equations

A B C D
Homework

Solutions for Session 6, Part D

See solutions for Problems: D1 | D2 | D3 | D4 | D5 | D6 | D7 | D8 | D9 | D10 | D11

Problem D1

The resulting graph is a line. It has a slope of 2, and passes through the points (0, -3), (1, -1), (2, 1), and many others.

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Problem D2

The equation for the line is y = 2x - 3.

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Problem D3

Pick two points: for example, (-1, -5) and (3, 3). The rise is the change in y, which is 3 - (-5) = 8. The run is the change in x, which is 3 - (-1) = 4. Then the slope, which is rise/run, is 8/4 = 2. Note that the slope, 2, is represented in the equation as the number that x is multiplied by.

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Problem D4

Yes, they share the point (2, 3).

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Problem D5

This point is the intersection of the lines, and the solution to the system of equations y = 2x - 1 and y = -x + 5. This also means that x = 2 is the solution to the balance equation 2x - 1 = -x + 5.

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Problem D6

The solution is the point (1, -3).

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Problem D7

The graphs intersect at the point (1, 2), which is the solution to the system. The solution to the balance equation 4x - 2 = 5x - 3 is x = 1.

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Problem D8

The graph is bowl-shaped, a parabola. Its lowest point is (0, 0). This point is the lowest because multiplying a number other than zero by itself always results in a positive number.

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Problem D9

The x-intercepts appear to be between x = 1 and 2, and between x = -2 and -1. A more exact way of solving the equation x² - 2 = 0 would be needed to find the exact values; the method of covering up works pretty well here. (Covered) - 2 = 0 leads to x² = 2, so the solutions are and its opposite. Rendered as decimals, these are approximately 1.414 and -1.414.

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Problem D10

The intersections are at the points (2, 2) and (-1, -1).

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Problem D11

Five donuts must cost $3.50, because this is the only thing that changes when Joe returns. This means that each donut costs 70 cents. Subtracting the cost of all the donuts from either trip tells us that 6 cups of coffee cost $6.60, so each cup of coffee costs $1.10.

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Session 6: Index | Notes | Solutions | Video