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# Checking Solutions to Equations

## Substitute values for variables to check equation solutions

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Checking Solutions to Equations

### Checking Solutions: Equations

You will often need to check solutions to equations in order to check your work. In a math class, verifying that you arrived at the correct solution is very good practice. We check a solution to an equation by replacing the variable in the equation with the value of the solution. A solution should result in a true statement when simplified.

#### Checking Solutions to Equations

1. Check that the given number is a solution to the equation: y=1; 3y+5=2y\begin{align*}y = -1; \ 3y + 5 = -2y\end{align*}

Replace the variable y\begin{align*}y\end{align*} in the equation with the given value, -1.

3(y)+53(1)+53+52=2(y)=2(1)=2=2\begin{align} 3(y)+5&=-2(y)\\ 3(-1) + 5 &= -2(-1)\\ -3 + 5 &= 2\\ 2 &= 2\\ \end{align}

2 = 2 is a true statement. This means that y=1\begin{align*}y = -1\end{align*} is a solution to 3y+5=2y\begin{align*}3y + 5 = -2y\end{align*}.

2. Check that the given number is a solution to the equation: z=3; z2+2z=8\begin{align*}z =3; \ z^2 + 2z = 8\end{align*}

Replace the variable z\begin{align*}z\end{align*} in the equation with the given value, 3.

(z)2+2(z)(3)2+2(3)9+615=8=8=8=8\begin{align} (z)^2+2(z)&=8\\ (3)^2 + 2(3) &= 8\\ 9 + 6 &= 8\\ 15 &= 8\\ \end{align}

15 = 8 is not a true statement. This means that z=3\begin{align*}z = 3\end{align*} is not a solution to z2+2z=8\begin{align*}z^2 + 2z = 8\end{align*} .

Let’s use what we have learned about defining variables, writing equations and writing inequalities to solve some real-world problems.

#### Real-World Applications

Tomatoes cost $0.50 each and avocados cost$2.00 each. Anne buys six more tomatoes than avocados. Her total bill is 8. How many tomatoes and how many avocados did Anne buy? Check your answer! Define Let a\begin{align*}a\end{align*} equal the number of avocados Anne buys. Translate Anne buys six more tomatoes than avocados. This means that a+6\begin{align*}a + 6\end{align*} is equal to the number of tomatoes. Tomatoes cost0.50 each and avocados cost $2.00 each. Her total bill is$8. This means that .50 times the number of tomatoes plus 2 times the number of avocados equals 8.

0.5(a+6)+2a0.5a+0.56+2a2.5a+32.5aa=8=8=8=5=2\begin{align} 0.5(a + 6) + 2a &= 8\\ 0.5a + 0.5 \cdot 6 + 2a &= 8\\ 2.5a + 3 &= 8\\ 2.5a &= 5\\ a &= 2\\ \end{align}

Remember that a\begin{align*}a\end{align*} represents the number of avocados. Since a=2\begin{align*}a=2\end{align*}, Anne buys two avocados. The number of tomatoes is (a)+6=(2)+6=8\begin{align*}(a) + 6 = (2) + 6 = 8\end{align*}.

Anne bought 2 avocados and 8 tomatoes.

Check

If Anne bought two avocados and eight tomatoes, the total cost would be: (2×2)+(8×0.5)=4+4=8\begin{align*}(2 \times 2) + (8 \times 0.5) = 4 + 4 = 8\end{align*}. The answer checks out.

### Example

#### Example 1

Check that the given number is a solution to the equation: x=12; 3x+1=x\begin{align*}x = -\frac{1}{2}; \ 3x + 1 = x\end{align*}

3(x)+13(12)+1(32)+112=x=12=12=12\begin{align} 3(x)+1&=x\\ 3 \left ( - \frac{1}{2} \right ) + 1 &= - \frac{1}{2}\\ \left ( - \frac{3}{2} \right ) + 1 &= - \frac{1}{2}\\ - \frac{1}{2} &= - \frac{1}{2}\\ \end{align}

12=12\begin{align*}-\frac{1}{2}=-\frac{1}{2}\end{align*} is a true statement. This means that x=12\begin{align*}x = - \frac{1}{2}\end{align*} is a solution to 3x+1=x\begin{align*}3x + 1 = x\end{align*}.

### Review

For 1-9, check whether the given number is a solution to the corresponding equation.

1. a=3; 4a+3=9\begin{align*}a = -3; \ 4a + 3 = -9\end{align*}
2. x=43; 34x+12=32\begin{align*}x = \frac{4}{3}; \ \frac{3}{4}x + \frac{1}{2} = \frac{3}{2}\end{align*}
3. y=2; 2.5y10.0=5.0\begin{align*}y = 2; \ 2.5y - 10.0 = -5.0\end{align*}
4. z=5; 2(52z)=202(z1)\begin{align*}z = -5; \ 2(5 - 2z) = 20 - 2(z - 1)\end{align*}
5. a=10; 5a7=43\begin{align*}a = 10; \ 5a-7=43\end{align*}
6. x=23;3x+5=7\begin{align*}x = \frac{2}{3}; 3x+5=7 \end{align*}
7. x=9; x310=30\begin{align*}x = -9; \ \frac{x}{3}\cdot 10 = -30\end{align*}
8. z=12; 2z=1.5z\begin{align*}z = \frac{1}{2}; \ 2z=1.5-z\end{align*}
9. z=0.5; z(12z)=6+6(4z3)\begin{align*}z = 0.5; \ z(1 - 2z) = 6 + 6(4z - 3)\end{align*}
10. The cost of a Ford Focus is 27% of the price of a Lexus GS 450h. If the price of the Ford is \$15000, what is the price of the Lexus?

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### Vocabulary Language: English

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.

solution

A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality.