Systems of Equations

To fully understand this Crash Course, you must be able to graph straight lines on a Cartesian plane.

It's also important to understand the rate of change (slope) and y-intercept of a linear function.

Graphing and Using a Table of Values

• Graphing can help determine the number of solutions of a system of equations. The diagram above can help you determine the number of solutions of a system of equations.

• A system of equations can also be solved using a table of values. To do so, you need to find the |x| value for which both |y| values are equal.

Algebraic Methods

• The comparison method allows you to accurately determine the solution pair when the same variable is isolated in both equations of the system of equations.
  Example: ||\begin{cases}y_1=5x-3\\y_2=2x+5\end{cases}\\[10pt]\begin{align}y_1&=y_2\\5x-3&=2x+5\\3x&=8\\x&=\dfrac{8}{3}\end{align}||

• The substitution method enables you to accurately determine the solution pair when one variable is isolated in one of the equations of the system of equations.
  Example: ||\begin{cases}-5y+8x=3\\y=2x-5\end{cases}\\[10pt]\begin{align}-5\boldsymbol{\color{#3a9a38}{y}}+8x&=3\\-5(\boldsymbol{\color{#3a9a38}{2x-5}})+8x&=3\\-10x+25+8x&=3\\-2x&=-22\\x&=11\end{align}||

• The elimination method allows you to accurately determine the solution pair when no variables are isolated in either of the 2 equations in the system and the 2 equations are written in the same form.
  Example: ||\begin{cases}2y+5x=26&\stackrel{\times3}{\rightarrow}\ 6y+15x=78\\3y-1x=5 &\stackrel{\times2}{\rightarrow}\ 6y-2x=10\end{cases}\\[10pt]\begin{alignat}{13}\cancel{6y}&+15x&&=78\\-\,(\cancel{6y}&-\ 2x&&=10)\\\hline&\quad17x&&=68\\&\!\qquad x&&=4\end{alignat}||

Note: A complete solution to a system of equations also requires calculating the value of the second variable by replacing the found value in one of the equations.

Lastly, don't forget that in a word problem, you must begin by defining the variables used in your equations.