Review A Chapter 7 - NOHS Teachers - Free Printable

Here are the step-by-step solutions for the problems on the worksheet.

Part 1: Is the ordered pair a solution?

To check if an ordered pair $(x, y)$ is a solution, plug the numbers into both equations. If both equations are true statements, it is a solution.

1. System: $x + y = 5$ and $-5x + 2y = -10$. Pair: $(0, 5)$.
* First equation: $0 + 5 = 5$. (True)
* Second equation: $-5(0) + 2(5) = 0 + 10 = 10$. The equation says it should equal $-10$. (False)
* Since one is false, it is not a solution.

2. System: $-x + y = -3$ and $x + 3y = 6$. Pair: $(4, 1)$.
* First equation: $-(4) + 1 = -3$. (True)
* Second equation: $4 + 3(1) = 4 + 3 = 7$. The equation says it should equal $6$. (False)
* It is not a solution.

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Part 2: Graph and Check

*(Note: Since I cannot draw on your paper, I will describe where the lines go so you can graph them.)*

3. System: $-x + y = 3$ and $x + y = 5$.
* Rewrite in slope-intercept form ($y = mx + b$):
* $y = x + 3$ (Start at 3 on the y-axis, go up 1, right 1).
* $y = -x + 5$ (Start at 5 on the y-axis, go down 1, right 1).
* They intersect at $(1, 4)$.

4. System: $x = 4$ and $y = 2$.
* $x = 4$ is a vertical line going through 4 on the x-axis.
* $y = 2$ is a horizontal line going through 2 on the y-axis.
* They intersect at $(4, 2)$.

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Part 3: Substitution Method

Substitute one variable into the other equation to solve.

5. System: $y = x + 2$ and $3x + 2y = 9$.
* Substitute $(x + 2)$ for $y$ in the second equation:
$$3x + 2(x + 2) = 9$$
$$3x + 2x + 4 = 9$$
$$5x + 4 = 9$$
$$5x = 5 \rightarrow x = 1$$
* Plug $x = 1$ back into the first equation:
$$y = 1 + 2 \rightarrow y = 3$$
* Solution: $(1, 3)$

6. System: $y = 3 - x$ and $-2x + y = 6$.
* Substitute $(3 - x)$ for $y$ in the second equation:
$$-2x + (3 - x) = 6$$
$$-3x + 3 = 6$$
$$-3x = 3 \rightarrow x = -1$$
* Plug $x = -1$ back into the first equation:
$$y = 3 - (-1) \rightarrow y = 4$$
* Solution: $(-1, 4)$

7. Word Problem: Student tickets ($s$) cost $\$3$, General tickets ($g$) cost $\$5$. Total tickets = 350. Total money = $\$1450$.
* Equation 1 (Count): $s + g = 350 \rightarrow s = 350 - g$
* Equation 2 (Money): $3s + 5g = 1450$
* Substitute $(350 - g)$ for $s$:
$$3(350 - g) + 5g = 1450$$
$$1050 - 3g + 5g = 1450$$
$$1050 + 2g = 1450$$
$$2g = 400 \rightarrow g = 200$$
* Find $s$: $s = 350 - 200 = 150$.
* Answer: 150 student tickets and 200 general admission tickets.

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Part 4: Linear Combinations (Elimination)

Add or subtract the equations to eliminate one variable.

8. System: $x + y = 5$ and $x - y = 3$.
* Add the two equations together:
$$(x + x) + (y - y) = 5 + 3$$
$$2x = 8 \rightarrow x = 4$$
* Plug $x = 4$ into the first equation:
$$4 + y = 5 \rightarrow y = 1$$
* Solution: $(4, 1)$

9. System: $x + y = 5$ and $2x + y = 6$.
* Subtract the top equation from the bottom equation:
$$(2x - x) + (y - y) = 6 - 5$$
$$x = 1$$
* Plug $x = 1$ into the first equation:
$$1 + y = 5 \rightarrow y = 4$$
* Solution: $(1, 4)$

10. Word Problem: Discs cost $\$11.50$ ($a$) and $\$7.50$ ($b$). Bought 12 discs. Spent $\$110$.
* Equation 1 (Count): $a + b = 12 \rightarrow b = 12 - a$
* Equation 2 (Cost): $11.50a + 7.50b = 110$
* To make math easier, multiply Eq 2 by 10 to remove decimals: $115a + 75b = 1100$. Divide by 5: $23a + 15b = 220$.
* Substitute $b = 12 - a$:
$$23a + 15(12 - a) = 220$$
$$23a + 180 - 15a = 220$$
$$8a + 180 = 220$$
$$8a = 40 \rightarrow a = 5$$
* Answer: You bought 5 compact discs that cost $\$11.50$.

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Part 5: Method of Choice & Number of Solutions

11. System: $x + 2y = 5$ and $2x - 2y = 4$.
* Use Elimination (add them because $+2y$ and $-2y$ cancel out):
$$(x + 2x) + (2y - 2y) = 5 + 4$$
$$3x = 9 \rightarrow x = 3$$
* Plug $x = 3$ into the first equation:
$$3 + 2y = 5$$
$$2y = 2 \rightarrow y = 1$$
* Solution: $(3, 1)$. The system has one solution.

12. System: $x + y = 1$ and $x + y = 3$.
* Look closely at the left sides. They are identical ($x+y$).
* However, the right sides are different ($1$ vs $3$).
* $x + y$ cannot equal 1 and 3 at the same time. The lines are parallel and never touch.
* Solution: No solution.

Final Answer:
1. No
2. No
3. (1, 4)
4. (4, 2)
5. (1, 3)
6. (-1, 4)
7. 150 student tickets, 200 general tickets
8. (4, 1)
9. (1, 4)
10. 5 discs
11. (3, 1); One solution
12. No solution