### Substitution Property of Equality

While the graphical approach to solving systems is helpful, it may not always provide exact answers. Therefore, consider a different method to solving systems, the **S****ubstitution Method**. This method uses the **Substitution Property of Equality**.

**Substitution Property of Equality **states that if \begin{align*}y=\end{align*} an algebraic expression, then the algebraic expression can be substituted for any \begin{align*}y\end{align*} in an equation or an inequality.

#### Let's find the solution to the system \begin{align*}\begin{cases} y=3x-5\\ y=-2x+5 \end{cases}\end{align*} using substitution:

Each equation is equal to the variable \begin{align*}y\end{align*}, therefore the two algebraic expressions must equal each other.

\begin{align*}3x-5=-2x+5\end{align*}

Solve for \begin{align*}x\end{align*}.

\begin{align*}3x-5+5&=-2x+5+5\\ 3x+2x&=-2x+2x+10\\ 5x&=10\\ x&=2\end{align*}

The \begin{align*}x\text{-}\end{align*}coordinate of the intersection of the two lines is 2. Now you must find the \begin{align*}y\text{-}\end{align*}coordinate using either of the two equations. Pick one of the original equations, and substitute 2 in place of \begin{align*}x:\end{align*}

\begin{align*}y=-2(2)+5=1\end{align*}

The solution to the system is \begin{align*}x=2, \ y=1\end{align*} or the ordered pair (2, 1).

#### Now, let's solve the following real-world problems using substitution:

- Peter and Nadia like to race each other. Peter can run at a speed of 5 feet per second and Nadia can run at a speed of 6 feet per second. To be a good sport, Nadia likes to give Peter a head start of 20 feet. How long does Nadia take to catch up with Peter? At what distance from the start does Nadia catch up with Peter?

The two racers' information can be translated into two equations.

Peter: \begin{align*}d=5t+20\end{align*}

Nadia: \begin{align*}d=6t\end{align*}

We want to know when the two racers will be the same distance from the start. This means we can set the two equations equal to each other.

\begin{align*}5t+20=6t\end{align*}

Now solve for \begin{align*}t\end{align*}.

\begin{align*}5t-5t+20& =6t-5t\\ 20& = 1t\end{align*}

After 20 seconds, Nadia will catch Peter.

Now we need to determine how far from the distance the two runners are. You already know \begin{align*}20=t\end{align*}, so we will substitute to determine the distance. Using either equation, substitute the known value for \begin{align*}t\end{align*} and find \begin{align*}d\end{align*}.

\begin{align*}d=5(20)+20 \rightarrow 120\end{align*}

When Nadia catches Peter, the runners are 120 feet from the starting line.

The substitution method is particularly useful when one equation in the system is of the form \begin{align*}y=\end{align*} algebraic expression or \begin{align*}x=\end{align*} algebraic expression.

- Anne is trying to choose between two phone plans. Vendaphone’s plan costs $20 per month, with calls costing an additional 25 cents per minute. Sellnet’s plan charges $40 per month, but calls cost only 8 cents per minute. Which should she choose?

Anne’s choice will depend upon how many minutes of calls she expects to use each month. We start by writing two equations for the cost in dollars in terms of the minutes used. Since the number of minutes is the independent variable, it will be our \begin{align*}x\end{align*}. Cost is dependent on minutes. The cost per month is the dependent variable and will be assigned \begin{align*}y\end{align*}.

\begin{align*}& \text{For Vendafone} && y =0.25x+20\\ & \text{For Sellnet} && y = 0.08x+40\end{align*}

By graphing two equations, we can see that at some point the two plans will charge the same amount, represented by the intersection of the two lines. Before this point, Sellnet’s plan is more expensive. After the intersection, Sellnet’s plan is cheaper.

Use substitution to find the point that the two plans are the same. Each algebraic expression is equal to \begin{align*}y\end{align*}, so they must equal each other.

\begin{align*}0.25x+20 & =0.08x+40 && \text{Subtract 20 from both sides.}\\ 0.25x & = 0.08x+20 && \text{Subtract} \ 0.08x \ \text{from both sides.}\\ 0.17x & = 20 && \text{Divide both sides by 0.17.}\\ x & = 117.65 \ \text{minutes} && \text{Rounded to two decimal places.}\end{align*}

We can now use our sketch, plus this information, to provide an answer. If Anne will use 117 minutes or fewer every month, she should choose Vendafone. If she plans on using 118 or more minutes, she should choose Sellnet.

### Examples

#### Example 1

Earlier, you were told that 4 bagels and 2 muffins cost $11 and 3 bagels and 3 muffins cost $12. If bagels and muffins cost different amounts, how much do individual bagels and muffins cost?

\begin{align*}\begin{cases} 4b+2m=11\\ 3b + 3m=12 \end{cases}\end{align*}

\begin{align*}3b + 3m &= 12\\ 3b &= 12-3m\\ b &=4-m\end{align*}

Now, we can substitute the expression in for \begin{align*}b\end{align*} in the first equation.

\begin{align*}4b + 2m &=11\\ 4(4-m) + 2m &= 11\\ 16 - 4m + 2m &= 11\\ 16- 2m &=11\\ -2m &= -5\\ m &= 2.5\end{align*}

Finally, we need to substitute the value for

The solution to the system is (1.5, 2.5). The bagels cost $1.50 and the muffins cost $2.50.

#### Example 2

Solve the system \begin{align*}\begin{cases} x+y=2\\ y=3 \end{cases}\end{align*}

The second equation is solved for the variable \begin{align*}y\end{align*}. Therefore, we can substitute the value “3” for any \begin{align*}y\end{align*} in the system.

\begin{align*}x+y=2 \rightarrow x+3=2\end{align*}

Now solve the equation for \begin{align*}x:\end{align*} \begin{align*}x+3-3& =2-3\\ x & = -1\end{align*}

The \begin{align*}x\text{-}\end{align*}coordinate of the intersection of these two equations is –1. Since we were given the \begin{align*}y\text{-}\end{align*}value of 3 in the question, we have our answer: \begin{align*}x=-1, \ y=3.\end{align*}

The solution to the system is (-1, 3), which you can verify by graphing the two lines.

### Review

- Explain the process of solving a system using the Substitution Property.
- Which systems are easier to solve using substitution?

Solve the following systems. Remember to find the value for both variables!

- \begin{align*}\begin{cases} y=\text{-}3\\ 6x-2y=0 \end{cases}\end{align*}
- \begin{align*}\ \\ \begin{cases} \text{-}3-3y=6\\ y=\text{-}3x+4 \end{cases}\\ \ \\\end{align*}
- \begin{align*}\begin{cases} y=3x+16\\ y=x+8 \end{cases}\end{align*}
- \begin{align*}\ \\ \begin{cases} y=\text{-}6x-3\\ y=3 \end{cases}\\ \ \\\end{align*}
- \begin{align*}\begin{cases} y=\text{-}2x+5\\ y=\text{-}1-8x \end{cases}\end{align*}
- \begin{align*}\ \\ \begin{cases} y=6+x\\ y=\text{-}2x-15 \end{cases}\\ \ \\\end{align*}
- \begin{align*}\begin{cases} y=\text{-}2\\ y=5x-17 \end{cases}\end{align*}
- \begin{align*}\ \\ \begin{cases} x+y=5\\ 3x+y=15 \end{cases}\\ \ \\\end{align*}
- \begin{align*}\begin{cases} 12y-3x=\text{-}1\\ x-4y=1 \end{cases}\end{align*}
- \begin{align*}\ \\ \begin{cases}x+2y=9\\ 3x+5y=20\\ \end{cases}\\ \ \\\end{align*}
- \begin{align*}\begin{cases}x-3y=10\\ 2x+y=13 \end{cases}\end{align*}
- Solve the system \begin{align*}\begin{cases} y=\frac{1}{4} x-14\\ y=\frac{19}{8} x+7 \end{cases}\end{align*} by both graphing and substitution. Which method do you prefer? Why?
- Of the two non-right angles in a right triangle, one measures twice that of the other. What are the angles?
- The sum of two numbers is 70. They differ by 11. What are the numbers?
- A rectangular field is enclosed by a fence on three sides and a wall on the fourth side. The total length of the fence is 320 yards. If the field has a total perimeter of 400 yards, what are the dimensions of the field?
- A ray cuts a line forming two angles. The difference between the two angles is \begin{align*}18^\circ\end{align*}. What does each angle measure?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.2.