Consider the following system of equations:

\begin{cases} x+ 3y = 6 \\ y = 9 \end{cases}

- How could you solve this without graphing?
- Would that method still work if the second equation was y=9-2x?

When at least one equation in a system of equations has an isolated variable (or a variable that can be easily isolated), the system can be solved efficiently by using a method called substitution.

The following steps can be used to solve a system of equations using substitution:

Steps | Example |
---|---|

\text{Given system} | x-y=5, 2x+3y=6 |

\text{1. Isolate a variable in one of the equations} | x-y=5 |

\text{ } | x=5+y |

\text{2. Substitute the resulting expression into the other equation} | 2\left(5+y\right)+3y=6 |

\text{3. Solve the equation for the variable} | 10+2y+3y=6 |

\text{ } | 10+5y=6 |

\text{ } | 5y=-4 |

\text{ } | y=-\dfrac{4}{5} |

\text{4. Substitute the value into one of the original equations} | x-\left(-\dfrac{4}{5}\right)=5 |

\text{5. Solve for the remaining variable} | x=\dfrac{21}{5} |

\text{6. Write the solution as an ordered pair} | \left(\dfrac{21}{5},-\dfrac{4}{5}\right) |

Recall that a **solution to a system of equations** is the set of ordered pairs that make all equations in the system true and that a system of linear equations can have three types of solutions:

When the solution to a system of equations does not consist of integer values it is difficult to determine the exact solution by graphing, so solving algebraically using a method like substitution is necessary.

Solve the following systems of equations using the substitution method.

a

\begin{cases}y=x+11\\y=3x+19 \end{cases}

Worked Solution

b

\begin{cases} 2x-3y=5 \\ x = 8 + 2y \end{cases}

Worked Solution

c

\begin{cases}y = \dfrac{1}{2}x\\ 4x-8y=20 \end{cases}

Worked Solution

The length of a rectangle is 3 inches less than twice its width. If the perimeter of the rectangle is 48 inches, find the length.

Worked Solution

A theater club at a high school charges a student rate and an adult rate to attend the spring musical. The cost for a student ticket is \$3 and the cost for an adult ticket is \$7. If 200 people attended the show and the theater club at the school raised \$700, determine how many students and how many adults attended.

Worked Solution

Idea summary

Solving systems of equations using the substitution method leads to one of three solutions:

- One solution: When solving algebraically an equation of the form x=a or y=b is reached
- No solutions: When solving algebraically an equation of the form a=b is reached
- Infinitely many solutions: When solving algebraically an equation of the form a=a is reached