# Systems of Equations Systems of Equations contains two or more equations with the same variables. Learn how to solve systems of equations.

## Step by step guide to solve systems of equations

• A system of equations contains two equations with similar two variables. For example, consider the system of equations: $$x -y=1, x+y=5$$
• The easiest way to solve a system of equation is using the elimination method. The elimination method uses the addition property of equality. You can add the same value to each side of an equation.
• For the first equation above, you can add $$x+y$$ to the left side and $$5$$ to the right side of the first equation: $$x-y+(x+y)=1+5$$. Now, if you simplify, you get: $$x-y+(x+y)=1+5→2x=6→x=3$$. Now, substitute $$3$$ for the $$x$$ in the first equation: $$3-y=1$$ . By solving this equation, $$y=2$$

### Example 1:

What is the value of $$x+y$$ in this system of equations? $$\begin{cases}2x+5y=11 \\ 4x-2y=-26\end{cases}$$

Solution:

Solving Systems of Equations by Elimination
Multiply the first equation by $$(−2)$$, then add it to the second equation.
\cfrac{\begin{align} -2(2x+5y=11) \\ 4x-2y=- \ 26 \end{align}}{}\Rightarrow \cfrac{\begin{align} -4x-10y=- \ 22 \\ 4x-2y=- \ 26 \end{align}}{}\Rightarrow -12y=-48\Rightarrow y=4
Plug in the value of $$y$$ into one of the equations and solve for $$x$$.
$$2x+5(4)= 11 ⇒ 2x+20= 11 ⇒ 2x=-9 ⇒ x=- \ 4.5$$
Thus, $$x+y=-4.5+4=- \ 0.5$$

### Example 2:

What is the value of $$x$$ and $$y$$ in this system of equations? $$\begin{cases}3x-4y= \ -20 \\ -x+2y=10\end{cases}$$

Solution:

Solving Systems of Equations by Elimination: $$\begin{cases}3x-4y= \ -20 \\ -x+2y=10\end{cases} \Rightarrow$$ Multiply the first equation by $$3$$, then add it to the second equation.
$$\begin{cases}3x-4y= \ -20 \\ 3(-x+2y=10)\end{cases} \Rightarrow \begin{cases}3x-4y= \ -20 \\ -3x+6y=30\end{cases} \Rightarrow 2y=10 \Rightarrow y=5$$
Now, substitute $$5$$ for $$y$$ in the first equation and solve for $$x. 3x−4(5)= −20→3x−20=−20→x=0$$

## Exercises

### Solve each system of equations.

1. $$\color{blue}{-4x-6y=7 \ \ \ \ \ x= \\ x-2y=7 \ \ \ \ \ y=} \\\$$
2. $$\color{blue}{-5x+y=-3 \ \ \ \ \ x= \\ 3x-7y=21 \ \ \ \ \ y= } \\\$$
3. $$\color{blue}{3y= -6x+12 \ \ \ \ \ x= \\ 8x-9y=-10 \ \ \ \ \ y= } \\\$$
4. $$\color{blue}{x+15y=50 \ \ \ \ \ x= \\ x+10y=40 \ \ \ \ \ y= } \\\$$
5. $$\color{blue}{3x-2y=15 \ \ \ \ \ x= \\ 3x-5y=15 \ \ \ \ \ y= } \\\$$
6. $$\color{blue}{3x-6y=-12 \ \ \ \ \ x= \\ -x-3y=-6 \ \ \ \ \ y= } \\\$$

1. $$\color{blue}{x=2,y=-\frac{5}{2}}$$
2. $$\color{blue}{x=0,y=-3}$$
3. $$\color{blue}{x=1,y=2}$$
4. $$\color{blue}{x=20,y=2}$$
5. $$\color{blue}{x=5,y=0}$$
6. $$\color{blue}{x=0,y=2}$$ 27% OFF

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