Question

solve each system by elimination. 11. \\(\

$$\begin{cases} -x + y = -1 \\\\ 2x - y = 0 \\end{cases}$$

\\) 12. \\(\

$$\begin{cases} -2x + y = -20 \\\\ 2x + y = 48 \\end{cases}$$

\\) 13. \\(\

$$\begin{cases} 3x - y = -2 \\\\ -2x + y = 3 \\end{cases}$$

\\) 14. \\(\

$$\begin{cases} x - y = 4 \\\\ x - 2y = 10 \\end{cases}$$

\\) 15. \\(\

$$\begin{cases} x + 2y = 5 \\\\ 3x + 2y = 17 \\end{cases}$$

\\) 16. \\(\

$$\begin{cases} 3x - 2y = -1 \\\\ 3x - 4y = 9 \\end{cases}$$

\\) 17. \\(\

$$\begin{cases} x - y = -3 \\\\ 5x + 3y = 1 \\end{cases}$$

\\) 18. \\(\

$$\begin{cases} 9x - 3y = 3 \\\\ 3x + 8y = -17 \\end{cases}$$

\\) 19. \\(\

$$\begin{cases} 5x + 2y = -1 \\\\ 3x + 7y = 11 \\end{cases}$$

\\) 20. multi-step mrs. gonzalez bought centerpieces to put on each table at a gradu party. she spent $31.50. there are 8 tables each requiring either a candle or vase candles cost $3 and vases cost $4.25. how many of each type did she buy?

Explanation:

Step1: Solve system 11 by adding equations

Add the two equations to eliminate $y$:
$$(-x + y) + (2x - y) = -1 + 0$$
$$x = -1$$
Substitute $x=-1$ into $-x + y = -1$:
$$-(-1) + y = -1 \implies 1 + y = -1 \implies y = -2$$

Step2: Solve system 12 by adding equations

Add the two equations to eliminate $x$:
$$(-2x + y) + (2x + y) = -20 + 48$$
$$2y = 28 \implies y = 14$$
Substitute $y=14$ into $2x + y = 48$:
$$2x + 14 = 48 \implies 2x = 34 \implies x = 17$$

Step3: Solve system13 by adding equations

Add the two equations to eliminate $y$:
$$(3x - y) + (-2x + y) = -2 + 3$$
$$x = 1$$
Substitute $x=1$ into $3x - y = -2$:
$$3(1) - y = -2 \implies 3 - y = -2 \implies y = 5$$

Step4: Solve system14 by subtracting equations

Subtract the second equation from the first to eliminate $x$:
$$(x - y) - (x - 2y) = 4 - 10$$
$$x - y - x + 2y = -6 \implies y = -6$$
Substitute $y=-6$ into $x - y = 4$:
$$x - (-6) = 4 \implies x + 6 = 4 \implies x = -2$$

Step5: Solve system15 by subtracting equations

Subtract the first equation from the second to eliminate $2y$:
$$(3x + 2y) - (x + 2y) = 17 - 5$$
$$2x = 12 \implies x = 6$$
Substitute $x=6$ into $x + 2y = 5$:
$$6 + 2y = 5 \implies 2y = -1 \implies y = -\frac{1}{2}$$

Step6: Solve system16 by subtracting equations

Subtract the first equation from the second to eliminate $3x$:
$$(3x - 4y) - (3x - 2y) = 9 - (-1)$$
$$3x - 4y - 3x + 2y = 10 \implies -2y = 10 \implies y = -5$$
Substitute $y=-5$ into $3x - 2y = -1$:
$$3x - 2(-5) = -1 \implies 3x + 10 = -1 \implies 3x = -11 \implies x = -\frac{11}{3}$$

Step7: Solve system17 by scaling first equation

Multiply first equation by 3, then add to the second:
$$3(x - y) = 3(-3) \implies 3x - 3y = -9$$
$$(3x - 3y) + (5x + 3y) = -9 + 1$$
$$8x = -8 \implies x = -1$$
Substitute $x=-1$ into $x - y = -3$:
$$-1 - y = -3 \implies -y = -2 \implies y = 2$$

Step8: Solve system18 by scaling second equation

Multiply second equation by 3, then subtract from first:
$$3(3x + 8y) = 3(-17) \implies 9x + 24y = -51$$
$$(9x - 3y) - (9x + 24y) = 3 - (-51)$$
$$9x - 3y - 9x -24y = 54 \implies -27y = 54 \implies y = -2$$
Substitute $y=-2$ into $3x + 8y = -17$:
$$3x + 8(-2) = -17 \implies 3x -16 = -17 \implies 3x = -1 \implies x = -\frac{1}{3}$$

Step9: Solve system19 by scaling equations

Multiply first equation by 3, second by 5, subtract:
$$3(5x + 2y) = 3(-1) \implies 15x + 6y = -3$$
$$5(3x + 7y) = 5(11) \implies 15x + 35y = 55$$
$$(15x + 6y) - (15x + 35y) = -3 - 55$$
$$-29y = -58 \implies y = 2$$
Substitute $y=2$ into $3x + 7y = 11$:
$$3x + 7(2) = 11 \implies 3x +14 = 11 \implies 3x = -3 \implies x = -1$$

Step10: Solve problem20, define variables

Let $c$ = number of candles, $v$ = number of vases. Set up system:

$$\begin{cases} c + v = 8 \\ 3c + 4.25v = 31.50 \end{cases}$$

Multiply first equation by 3, subtract from second:
$$3(c + v) = 3(8) \implies 3c + 3v = 24$$
$$(3c + 4.25v) - (3c + 3v) = 31.50 -24$$
$$1.25v = 7.50 \implies v = 6$$
Substitute $v=6$ into $c + v =8$:
$$c +6 =8 \implies c=2$$

Answer:

1. $x=-1,\ y=-2$
2. $x=17,\ y=14$
3. $x=1,\ y=5$
4. $x=-2,\ y=-6$
5. $x=6,\ y=-\frac{1}{2}$
6. $x=-\frac{11}{3},\ y=-5$
7. $x=-1,\ y=2$
8. $x=-\frac{1}{3},\ y=-2$
9. $x=-1,\ y=2$
10. 2 candles and 6 vases