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1.8: Checking Solutions to Equations

Difficulty Level: At Grade Created by: CK-12
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Practice Checking Solutions to Equations

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What if you were given an equation like \begin{align*}2x^2 - 8 = 0\end{align*} and told that one of its solutions was \begin{align*}x = -2\end{align*}? How could you determine if that solution were correct? After completing this Concept, you'll be able to check the solutions to equations like this one.

Guidance

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

Example A

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

Solution

Replace the variable in each equation with the given value.

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

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

Example B

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

Solution:

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

This is not a true statement. This means that \begin{align*}z = 3\end{align*} is not a solution to \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.

Example C

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! Solution Define Let \begin{align*}a = \end{align*} the number of avocados Anne buys. Translate Anne buys six more tomatoes than avocados. This means that \begin{align*}a + 6 =\end{align*} 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.

\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 \begin{align*}a =\end{align*} the number of avocados, so Anne buys two avocados. The number of tomatoes is \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 is: \begin{align*}(2 \times 2) + (8 \times 0.5) = 4 + 4 = 8\end{align*}. The answer checks out.

Watch this video for help with the Examples above.

Vocabulary

• A solution an equation should result in a true statement when plugged into the equation.

Guided Practice

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

Solution:

\begin{align*} 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*}

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

Practice

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

1. \begin{align*}a = -3; \ 4a + 3 = -9\end{align*}
2. \begin{align*}x = \frac{4}{3}; \ \frac{3}{4}x + \frac{1}{2} = \frac{3}{2}\end{align*}
3. \begin{align*}y = 2; \ 2.5y - 10.0 = -5.0\end{align*}
4. \begin{align*}z = -5; \ 2(5 - 2z) = 20 - 2(z - 1)\end{align*}
5. \begin{align*}a = 10; \ 5a-7=43\end{align*}
6. \begin{align*}x = \frac{2}{3}; 3x+5=7 \end{align*}
7. \begin{align*}y = -9; \ \frac{x}{3}\cdot 10 = -30\end{align*}
8. \begin{align*}z = \frac{1}{2}; \ 2z=1.5-z\end{align*}
9. \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

TermDefinition
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.

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