How to Solve Systems of Equations? (+FREE Worksheet!)
Systems of Equations contain two or more equations with the same variables. Learn how to solve systems of equations.
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Step-by-step guide to solving 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 equations 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, \(y=2\)
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Systems of Equations – 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 \(x\).
\(2x+5y=11\) ⇒ \(2x+5(4)= 11 ⇒ 2x+20= 11 ⇒ 2x=-9 ⇒ x=- \ 4.5\)
Thus, \(x+y=-4.5+4=- \ 0.5\)
Systems of Equations – 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 \(x\):
\(3x-4y=-20 → 3x−4(5)= −20→3x−20=−20→x=0\)
Exercises for Solving Systems of Equations
Solve each system of equations.
- \(\color{blue}{-4x-6y=7 \ \ \ \ \ x= \\ x-2y=7 \ \ \ \ \ y=} \\\ \)
- \(\color{blue}{-5x+y=-3 \ \ \ \ \ x= \\ 3x-7y=21 \ \ \ \ \ y= } \\\ \)
- \(\color{blue}{3y= -6x+12 \ \ \ \ \ x= \\ 8x-9y=-10 \ \ \ \ \ y= } \\\ \)
- \(\color{blue}{x+15y=50 \ \ \ \ \ x= \\ x+10y=40 \ \ \ \ \ y= } \\\ \)
- \(\color{blue}{3x-2y=15 \ \ \ \ \ x= \\ 3x-5y=15 \ \ \ \ \ y= } \\\ \)
- \(\color{blue}{3x-6y=-12 \ \ \ \ \ x= \\ -x-3y=-6 \ \ \ \ \ y= } \\\ \)
Download the Systems of Equations Worksheet
Answers
- \(\color{blue}{x=2,y=-\frac{5}{2}}\)
- \(\color{blue}{x=0,y=-3}\)
- \(\color{blue}{x=1,y=2}\)
- \(\color{blue}{x=20,y=2}\)
- \(\color{blue}{x=5,y=0}\)
- \(\color{blue}{x=0,y=2}\)
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