How to Solve Systems of Equations

How to Solve Systems of Equations

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

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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, \(y=2\)

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 \(x\).
\(2x+5(4)= 11 ⇒ 2x+20= 11 ⇒ 2x=-9 ⇒ x=- \ 4.5\)
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 \(x. 3x−4(5)= −20→3x−20=−20→x=0\)

Exercises for Solving Systems of Equations

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. \(\color{blue}{3x-6y=-12 \ \ \ \ \ x= \\ -x-3y=-6 \ \ \ \ \ y= } \\\ \)

Download Systems of Equations Worksheet

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Answers

  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. \(\color{blue}{x=0,y=2}\)

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