Question

solving systems using elimination
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period
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solve each system of equations using the elimination method. check the solution.

1. $x - 2y = 10$

$- x + 3y = 5$

1. $x + y = -4$

$x - y = 2$

1. $3x + 4y = 23$

$3x - 2y = 11$

1. $10x - 6y = 54$

$-4x + 3y = -24$

1. $4x + 2y = 6$

$4x - 3y = -19$

1. $4x - y = -2$

$2x + y = 2$

1. $3x + 8y = 2$

$4x - 2y = -10$

1. $2x - 4y = 10$

$-4x + 3y = -20$

Explanation:

Step1: Solve system 2

We have the system:

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

Add the two equations to eliminate \(y\):
\((x + y)+(x - y)=-4 + 2\)
\(2x=-2\)

Step2: Find \(x\)

Divide both sides by 2:
\(x=\frac{-2}{2}=-1\)

Step3: Find \(y\)

Substitute \(x = - 1\) into \(x + y=-4\):
\(-1+y=-4\)
Add 1 to both sides:
\(y=-4 + 1=-3\)

Step4: Check the solution

Substitute \(x=-1,y = - 3\) into the first equation: \(x + y=-1+(-3)=-4\) (matches).
Substitute into the second equation: \(x - y=-1-(-3)=-1 + 3 = 2\) (matches).

Step1: Solve system 3

We have the system:

$$\begin{cases} 3x + 4y=23 \\ 3x-2y = 11 \end{cases}$$

Subtract the second equation from the first to eliminate \(x\):
\((3x + 4y)-(3x - 2y)=23-11\)
\(3x+4y - 3x + 2y=12\)
\(6y=12\)

Step2: Find \(y\)

Divide both sides by 6:
\(y = \frac{12}{6}=2\)

Step3: Find \(x\)

Substitute \(y = 2\) into \(3x-2y=11\):
\(3x-2\times2=11\)
\(3x-4 = 11\)
Add 4 to both sides: \(3x=11 + 4=15\)
Divide by 3: \(x=\frac{15}{3}=5\)

Step4: Check the solution

Substitute \(x = 5,y=2\) into \(3x + 4y\): \(3\times5+4\times2=15 + 8=23\) (matches).
Substitute into \(3x-2y\): \(3\times5-2\times2=15 - 4 = 11\) (matches).

Step1: Solve system 4

We have the system:

$$\begin{cases} 10x-6y=54 \\ -4x + 3y=-24 \end{cases}$$

Multiply the second equation by 2: \(-8x + 6y=-48\)
Now add this new equation to the first equation:
\((10x-6y)+(-8x + 6y)=54+(-48)\)
\(10x-6y-8x + 6y=6\)
\(2x=6\)

Step2: Find \(x\)

Divide by 2: \(x = 3\)

Step3: Find \(y\)

Substitute \(x = 3\) into \(10x-6y=54\):
\(10\times3-6y=54\)
\(30-6y=54\)
Subtract 30 from both sides: \(-6y=54 - 30=24\)
Divide by - 6: \(y=\frac{24}{-6}=-4\)

Step4: Check the solution

Substitute \(x = 3,y=-4\) into \(10x-6y\): \(10\times3-6\times(-4)=30 + 24=54\) (matches).
Substitute into \(-4x + 3y\): \(-4\times3+3\times(-4)=-12-12=-24\) (matches).

Step1: Solve system 6

We have the system:

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

Add the two equations to eliminate \(y\):
\((4x-y)+(2x + y)=-2 + 2\)
\(4x-y+2x + y=0\)
\(6x=0\)

Step2: Find \(x\)

Divide by 6: \(x = 0\)

Step3: Find \(y\)

Substitute \(x = 0\) into \(2x + y=2\):
\(0+y=2\)
\(y = 2\)

Step4: Check the solution

Substitute \(x = 0,y = 2\) into \(4x-y\): \(4\times0-2=-2\) (matches).
Substitute into \(2x + y\): \(2\times0+2=2\) (matches).

Step1: Solve system 7

We have the system:

$$\begin{cases} 3x + 8y=2 \\ 4x-2y=-10 \end{cases}$$

Multiply the second equation by 4: \(16x-8y=-40\)
Add this new equation to the first equation:
\((3x + 8y)+(16x-8y)=2+(-40)\)
\(3x + 8y+16x-8y=-38\)
\(19x=-38\)

Step2: Find \(x\)

Divide by 19: \(x=\frac{-38}{19}=-2\)

Step3: Find \(y\)

Substitute \(x=-2\) into \(3x + 8y=2\):
\(3\times(-2)+8y=2\)
\(-6 + 8y=2\)
Add 6 to both sides: \(8y=2 + 6=8\)
Divide by 8: \(y = 1\)

Step4: Check the solution

Substitute \(x=-2,y = 1\) into \(3x + 8y\): \(3\times(-2)+8\times1=-6 + 8=2\) (matches).
Substitute into \(4x-2y\): \(4\times(-2)-2\times1=-8-2=-10\) (matches).

Step1: Solve system 8

We have the system:

$$\begin{cases} 2x-4y=10 \\ -4x + 3y=-20 \end{cases}$$

Multiply the first equation by 2: \(4x-8y=20\)
Add this new equation to the second equation:
\((4x-8y)+(-4x + 3y)=20+(-20)\)
\(4x-8y-4x + 3y=0\)
\(-5y=0\)

Step2: Find \(y\)

Divide by - 5: \(y = 0\)

Step3: Find \(x\)

Substitute \(y = 0\) into \(2x-4y=10\):
\(2x-4\times0=10\)
\(2x=10\)
Divide by 2: \(x = 5\)

Step4: Check the solution

Substitute \(x = 5,y = 0\) into \(2x-4y\): \(2\times5-4\times0=10\) (matches).
Substitute into \(-4x + 3y\): \(-4\times5…

Answer:

1. Solution: \((8,-1)\)
2. Solution: \((-1,-3)\)
3. Solution: \((5,2)\)
4. Solution: \((3,-4)\)
5. Solution: \((-5,13)\)
6. Solution: \((0,2)\)
7. Solution: \((-2,1)\)
8. Solution: \((5,0)\)