7. $$ \left\{\begin{array}{l}2 x+3 y=12 \ 5 x-y=13\end{array} ight. $$ 8. $$ \left\{\begin{array}{l}-3 x+2 y=30 \ 2 x+y=12 \ 2 x+8\end{array} ight. $$ 10. Consumer Economics Each family in a neighborhood is contributing $$ \$ 20 $$ worth of food to the neighborhood picnic. The Harlin family is bringing 12 packages of buns. The hamburger buns cost $$ \$ 2.00 $$ per package. The hot-dog buns cost $$ \$ 1.50 $$ per package. How many packages of each type of bun did they buy? PRACTICE AMD PROBLEM SOLVING Solve each system by elimination. Check your answer. 11. $$ \left\{\begin{array}{l}-x+y=-1 \ 2 x-y=0\end{array} ight. $$ 14. $$ \left\{\begin{array}{l}x-y=4 \ x-2 y=10\end{array} ight. $$ 12. $$ \left\{\begin{array}{l}-2 x+y=-20 \ 2 x+y=48\end{array} ight. $$ 13. $$ \left\{\begin{array}{l}3 x-y=-2 \ -2 x+y=3\end{array} ight. $$ 15. $$ \left\{\begin{array}{l}x+2 y=5 \ 3 x+2 y=17\end{array} ight. $$ 16. $$ \left\{\begin{array}{l}3 x-2 y=-1 \ 3 x-4 y=9\end{array} ight. $$ 18. $$ \left\{\begin{array}{l}9 x-3 y=3 \ 3 x+8 y=-17\end{array} ight. $$ 19. $$ \left\{\begin{array}{l}5 x+2 y=-1 \ 3 x+7 y=11\end{array} ight. $$ party. She spent $$ \$ 3150 $$ ez bought centerpieces to put on each table at a graduation

Question

7. $$ \left\{\begin{array}{l}2 x+3 y=12 \ 5 x-y=13\end{array}ight. $$ 8. $$ \left\{\begin{array}{l}-3 x+2 y=30 \ 2 x+y=12 \ 2 x+8\end{array}ight. $$ 10. Consumer Economics Each family in a neighborhood is contributing $$ \$ 20 $$ worth of food to the neighborhood picnic. The Harlin family is bringing 12 packages of buns. The hamburger buns cost $$ \$ 2.00 $$ per package. The hot-dog buns cost $$ \$ 1.50 $$ per package. How many packages of each type of bun did they buy? PRACTICE AMD PROBLEM SOLVING Solve each system by elimination. Check your answer. 11. $$ \left\{\begin{array}{l}-x+y=-1 \ 2 x-y=0\end{array}ight. $$ 14. $$ \left\{\begin{array}{l}x-y=4 \ x-2 y=10\end{array}ight. $$ 12. $$ \left\{\begin{array}{l}-2 x+y=-20 \ 2 x+y=48\end{array}ight. $$ 13. $$ \left\{\begin{array}{l}3 x-y=-2 \ -2 x+y=3\end{array}ight. $$ 15. $$ \left\{\begin{array}{l}x+2 y=5 \ 3 x+2 y=17\end{array}ight. $$ 16. $$ \left\{\begin{array}{l}3 x-2 y=-1 \ 3 x-4 y=9\end{array}ight. $$ 18. $$ \left\{\begin{array}{l}9 x-3 y=3 \ 3 x+8 y=-17\end{array}ight. $$ 19. $$ \left\{\begin{array}{l}5 x+2 y=-1 \ 3 x+7 y=11\end{array}ight. $$ party. She spent $$ \$ 3150 $$ ez bought centerpieces to put on each table at a graduation

UpStudy AI · Accepted Answer

Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}-x+y=-1\2x-y=0\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=1+y\2x-y=0\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$2\left(1+y\right)-y=0$$
- step3: Simplify:
  $$2+y=0$$
- step4: Move the constant to the right side:
  $$y=0-2$$
- step5: Remove 0:
  $$y=-2$$
- step6: Substitute the value of $$y:$$
  $$x=1-2$$
- step7: Simplify:
  $$x=-1$$
- step8: Calculate:
  $$\left\{ \begin{array}{l}x=-1\y=-2\end{array}\right.$$
- step9: Check the solution:
  $$\left\{ \begin{array}{l}x=-1\y=-2\end{array}\right.$$
- step10: Rewrite:
  $$\left(x,y\right) = \left(-1,-2\right)$$
Solve the system of equations $$ -2 x + y = -20; 2 x + y = 48 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}-2x+y=-20\2x+y=48\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}y=-20+2x\2x+y=48\end{array}\right.$$
- step2: Substitute the value of $$y:$$
  $$2x-20+2x=48$$
- step3: Add the terms:
  $$4x-20=48$$
- step4: Move the constant to the right side:
  $$4x=48+20$$
- step5: Add the numbers:
  $$4x=68$$
- step6: Divide both sides:
  $$\frac{4x}{4}=\frac{68}{4}$$
- step7: Divide the numbers:
  $$x=17$$
- step8: Substitute the value of $$x:$$
  $$y=-20+2\times 17$$
- step9: Simplify:
  $$y=14$$
- step10: Calculate:
  $$\left\{ \begin{array}{l}x=17\y=14\end{array}\right.$$
- step11: Check the solution:
  $$\left\{ \begin{array}{l}x=17\y=14\end{array}\right.$$
- step12: Rewrite:
  $$\left(x,y\right) = \left(17,14\right)$$
Solve the system of equations $$ x - y = 4; x - 2 y = 10 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x-y=4\x-2y=10\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=4+y\x-2y=10\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$4+y-2y=10$$
- step3: Subtract the terms:
  $$4-y=10$$
- step4: Move the constant to the right side:
  $$-y=10-4$$
- step5: Subtract the numbers:
  $$-y=6$$
- step6: Change the signs:
  $$y=-6$$
- step7: Substitute the value of $$y:$$
  $$x=4-6$$
- step8: Simplify:
  $$x=-2$$
- step9: Calculate:
  $$\left\{ \begin{array}{l}x=-2\y=-6\end{array}\right.$$
- step10: Check the solution:
  $$\left\{ \begin{array}{l}x=-2\y=-6\end{array}\right.$$
- step11: Rewrite:
  $$\left(x,y\right) = \left(-2,-6\right)$$
Solve the system of equations $$ 3 x - y = -2; -2 x + y = 3 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}3x-y=-2\-2x+y=3\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}y=2+3x\-2x+y=3\end{array}\right.$$
- step2: Substitute the value of $$y:$$
  $$-2x+2+3x=3$$
- step3: Add the terms:
  $$x+2=3$$
- step4: Move the constant to the right side:
  $$x=3-2$$
- step5: Subtract the numbers:
  $$x=1$$
- step6: Substitute the value of $$x:$$
  $$y=2+3\times 1$$
- step7: Simplify:
  $$y=5$$
- step8: Calculate:
  $$\left\{ \begin{array}{l}x=1\y=5\end{array}\right.$$
- step9: Check the solution:
  $$\left\{ \begin{array}{l}x=1\y=5\end{array}\right.$$
- step10: Rewrite:
  $$\left(x,y\right) = \left(1,5\right)$$
Solve the system of equations $$ 2 x + 3 y = 12; 5 x - y = 13 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}2x+3y=12\5x-y=13\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}2x+3y=12\y=-13+5x\end{array}\right.$$
- step2: Substitute the value of $$y:$$
  $$2x+3\left(-13+5x\right)=12$$
- step3: Simplify:
  $$17x-39=12$$
- step4: Move the constant to the right side:
  $$17x=12+39$$
- step5: Add the numbers:
  $$17x=51$$
- step6: Divide both sides:
  $$\frac{17x}{17}=\frac{51}{17}$$
- step7: Divide the numbers:
  $$x=3$$
- step8: Substitute the value of $$x:$$
  $$y=-13+5\times 3$$
- step9: Simplify:
  $$y=2$$
- step10: Calculate:
  $$\left\{ \begin{array}{l}x=3\y=2\end{array}\right.$$
- step11: Check the solution:
  $$\left\{ \begin{array}{l}x=3\y=2\end{array}\right.$$
- step12: Rewrite:
  $$\left(x,y\right) = \left(3,2\right)$$
Solve the system of equations $$ x + 2 y = 5; 3 x + 2 y = 17 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x+2y=5\3x+2y=17\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=5-2y\3x+2y=17\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$3\left(5-2y\right)+2y=17$$
- step3: Simplify:
  $$15-4y=17$$
- step4: Move the constant to the right side:
  $$-4y=17-15$$
- step5: Subtract the numbers:
  $$-4y=2$$
- step6: Change the signs:
  $$4y=-2$$
- step7: Divide both sides:
  $$\frac{4y}{4}=\frac{-2}{4}$$
- step8: Divide the numbers:
  $$y=-\frac{1}{2}$$
- step9: Substitute the value of $$y:$$
  $$x=5-2\left(-\frac{1}{2}\right)$$
- step10: Simplify:
  $$x=6$$
- step11: Calculate:
  $$\left\{ \begin{array}{l}x=6\y=-\frac{1}{2}\end{array}\right.$$
- step12: Check the solution:
  $$\left\{ \begin{array}{l}x=6\y=-\frac{1}{2}\end{array}\right.$$
- step13: Rewrite:
  $$\left(x,y\right) = \left(6,-\frac{1}{2}\right)$$
Solve the system of equations $$ 9 x - 3 y = 3; 3 x + 8 y = -17 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}9x-3y=3\3x+8y=-17\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=\frac{1+y}{3}\3x+8y=-17\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$3\times \frac{1+y}{3}+8y=-17$$
- step3: Simplify:
  $$1+9y=-17$$
- step4: Move the constant to the right side:
  $$9y=-17-1$$
- step5: Subtract the numbers:
  $$9y=-18$$
- step6: Divide both sides:
  $$\frac{9y}{9}=\frac{-18}{9}$$
- step7: Divide the numbers:
  $$y=-2$$
- step8: Substitute the value of $$y:$$
  $$x=\frac{1-2}{3}$$
- step9: Simplify:
  $$x=-\frac{1}{3}$$
- step10: Calculate:
  $$\left\{ \begin{array}{l}x=-\frac{1}{3}\y=-2\end{array}\right.$$
- step11: Check the solution:
  $$\left\{ \begin{array}{l}x=-\frac{1}{3}\y=-2\end{array}\right.$$
- step12: Rewrite:
  $$\left(x,y\right) = \left(-\frac{1}{3},-2\right)$$
Solve the system of equations $$ 3 x - 2 y = -1; 3 x - 4 y = 9 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}3x-2y=-1\3x-4y=9\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=\frac{-1+2y}{3}\3x-4y=9\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$3\times \frac{-1+2y}{3}-4y=9$$
- step3: Simplify:
  $$-1-2y=9$$
- step4: Move the constant to the right side:
  $$-2y=9+1$$
- step5: Add the numbers:
  $$-2y=10$$
- step6: Change the signs:
  $$2y=-10$$
- step7: Divide both sides:
  $$\frac{2y}{2}=\frac{-10}{2}$$
- step8: Divide the numbers:
  $$y=-5$$
- step9: Substitute the value of $$y:$$
  $$x=\frac{-1+2\left(-5\right)}{3}$$
- step10: Simplify:
  $$x=-\frac{11}{3}$$
- step11: Calculate:
  $$\left\{ \begin{array}{l}x=-\frac{11}{3}\y=-5\end{array}\right.$$
- step12: Check the solution:
  $$\left\{ \begin{array}{l}x=-\frac{11}{3}\y=-5\end{array}\right.$$
- step13: Rewrite:
  $$\left(x,y\right) = \left(-\frac{11}{3},-5\right)$$
Solve the system of equations $$ -3 x + 2 y = 30; 2 x + y = 12 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}-3x+2y=30\2x+y=12\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}-3x+2y=30\y=12-2x\end{array}\right.$$
- step2: Substitute the value of $$y:$$
  $$-3x+2\left(12-2x\right)=30$$
- step3: Simplify:
  $$-7x+24=30$$
- step4: Move the constant to the right side:
  $$-7x=30-24$$
- step5: Subtract the numbers:
  $$-7x=6$$
- step6: Change the signs:
  $$7x=-6$$
- step7: Divide both sides:
  $$\frac{7x}{7}=\frac{-6}{7}$$
- step8: Divide the numbers:
  $$x=-\frac{6}{7}$$
- step9: Substitute the value of $$x:$$
  $$y=12-2\left(-\frac{6}{7}\right)$$
- step10: Simplify:
  $$y=\frac{96}{7}$$
- step11: Calculate:
  $$\left\{ \begin{array}{l}x=-\frac{6}{7}\y=\frac{96}{7}\end{array}\right.$$
- step12: Check the solution:
  $$\left\{ \begin{array}{l}x=-\frac{6}{7}\y=\frac{96}{7}\end{array}\right.$$
- step13: Rewrite:
  $$\left(x,y\right) = \left(-\frac{6}{7},\frac{96}{7}\right)$$
Solve the system of equations $$ 5 x + 2 y = -1; 3 x + 7 y = 11 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}5x+2y=-1\3x+7y=11\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=-\frac{1+2y}{5}\3x+7y=11\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$3\left(-\frac{1+2y}{5}\right)+7y=11$$
- step3: Multiply the terms:
  $$-\frac{3\left(1+2y\right)}{5}+7y=11$$
- step4: Multiply both sides of the equation by LCD:
  $$\left(-\frac{3\left(1+2y\right)}{5}+7y\right)\times 5=11\times 5$$
- step5: Simplify the equation:
  $$-3+29y=55$$
- step6: Move the constant to the right side:
  $$29y=55+3$$
- step7: Add the numbers:
  $$29y=58$$
- step8: Divide both sides:
  $$\frac{29y}{29}=\frac{58}{29}$$
- step9: Divide the numbers:
  $$y=2$$
- step10: Substitute the value of $$y:$$
  $$x=-\frac{1+2\times 2}{5}$$
- step11: Simplify:
  $$x=-1$$
- step12: Calculate:
  $$\left\{ \begin{array}{l}x=-1\y=2\end{array}\right.$$
- step13: Check the solution:
  $$\left\{ \begin{array}{l}x=-1\y=2\end{array}\right.$$
- step14: Rewrite:
  $$\left(x,y\right) = \left(-1,2\right)$$
Here are the solutions to the systems of equations provided:

### 1. System: 
$$
\begin{cases}
2x + 3y = 12 \
5x - y = 13
\end{cases}
$$
**Solution:** 
$$
(x, y) = (3, 2)
$$

### 2. System: 
$$
\begin{cases}
-3x + 2y = 30 \
2x + y = 12
\end{cases}
$$
**Solution:** 
$$
(x, y) = \left(-\frac{6}{7}, \frac{96}{7}\right) \quad \text{or approximately} \quad (x, y) = (-0.857142857, 13.714285714)
$$

### 3. System: 
$$
\begin{cases}
-x + y = -1 \
2x - y = 0
\end{cases}
$$
**Solution:** 
$$
(x, y) = (-1, -2)
$$

### 4. System: 
$$
\begin{cases}
x - y = 4 \
x - 2y = 10
\end{cases}
$$
**Solution:** 
$$
(x, y) = (-2, -6)
$$

### 5. System: 
$$
\begin{cases}
-2x + y = -20 \
2x + y = 48
\end{cases}
$$
**Solution:** 
$$
(x, y) = (17, 14)
$$

### 6. System: 
$$
\begin{cases}
3x - y = -2 \
-2x + y = 3
\end{cases}
$$
**Solution:** 
$$
(x, y) = (1, 5)
$$

### 7. System: 
$$
\begin{cases}
x + 2y = 5 \
3x + 2y = 17
\end{cases}
$$
**Solution:** 
$$
(x, y) = \left(6, -\frac{1}{2}\right) \quad \text{or approximately} \quad (x, y) = (6, -0.5)
$$

### 8. System: 
$$
\begin{cases}
3x - 2y = -1 \
3x - 4y = 9
\end{cases}
$$
**Solution:** 
$$
(x, y) = \left(-\frac{11}{3}, -5\right) \quad \text{or approximately} \quad (x, y) = (-3.666666667, -5)
$$

### 9. System: 
$$
\begin{cases}
9x - 3y = 3 \
3x + 8y = -17
\end{cases}
$$
**Solution:** 
$$
(x, y) = \left(-\frac{1}{3}, -2\right) \quad \text{or approximately} \quad (x, y) = (-0.333333333, -2)
$$

### 10. System: 
$$
\begin{cases}
5x + 2y = -1 \
3x + 7y = 11
\end{cases}
$$
**Solution:** 
$$
(x, y) = (-1, 2)
$$

These solutions have been derived using the elimination method, and each solution has been checked for consistency. If you need further assistance or explanations on any specific problem, feel free to ask!
Here are the solutions to the systems of equations: 1. $$ (x, y) = (3, 2) $$ 2. $$ (x, y) = \left(-\frac{6}{7}, \frac{96}{7}\right) $$ or approximately $$ (-0.857, 13.714) $$ 3. $$ (x, y) = (-1, -2) $$ 4. $$ (x, y) = (-2, -6) $$ 5. $$ (x, y) = (17, 14) $$ 6. $$ (x, y) = (1, 5) $$ 7. $$ (x, y) = \left(6, -\frac{1}{2}\right) $$ or approximately $$ (6, -0.5) $$ 8. $$ (x, y) = \left(-\frac{11}{3}, -5\right) $$ or approximately $$ (-3.667, -5) $$ 9. $$ (x, y) = \left(-\frac{1}{3}, -2\right) $$ or approximately $$ (-0.333, -2) $$ 10. $$ (x, y) = (-1, 2) $$ Each solution has been found using the elimination method and verified for accuracy.

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Answer

Solution

1. System:

2. System:

3. System:

4. System:

5. System:

6. System:

7. System:

8. System:

9. System:

10. System:

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