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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: \begin{align*}y = -1; \ 3y + 5 = -2y\end{align*}y=1; 3y+5=2y

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

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

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

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

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

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

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

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 \begin{align*}a\end{align*}a equal the number of avocados Anne buys.

Translate

Anne buys six more tomatoes than avocados. This means that \begin{align*}a + 6\end{align*}a+6 is equal to the number of tomatoes.

Tomatoes cost $0.50 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}0.5(a+6)+2a0.5a+0.56+2a2.5a+32.5aa=8=8=8=5=2

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

Answer

Anne bought 2 avocados and 8 tomatoes.

Check

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

Example

Example 1

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

\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}3(x)+13(12)+1(32)+112=x=12=12=12

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

Review

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*}a=3; 4a+3=9
  2. \begin{align*}x = \frac{4}{3}; \ \frac{3}{4}x + \frac{1}{2} = \frac{3}{2}\end{align*}x=43; 34x+12=32
  3. \begin{align*}y = 2; \ 2.5y - 10.0 = -5.0\end{align*}y=2; 2.5y10.0=5.0
  4. \begin{align*}z = -5; \ 2(5 - 2z) = 20 - 2(z - 1)\end{align*}z=5; 2(52z)=202(z1)
  5. \begin{align*}a = 10; \ 5a-7=43\end{align*}a=10; 5a7=43
  6. \begin{align*}x = \frac{2}{3}; 3x+5=7 \end{align*}x=23;3x+5=7
  7. \begin{align*}x = -9; \ \frac{x}{3}\cdot 10 = -30\end{align*}x=9; x310=30
  8. \begin{align*}z = \frac{1}{2}; \ 2z=1.5-z\end{align*}z=12; 2z=1.5z
  9. \begin{align*}z = 0.5; \ z(1 - 2z) = 6 + 6(4z - 3)\end{align*}z=0.5; z(12z)=6+6(4z3)
  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?

Review (Answers)

To view the Review answers, open this PDF file and look for section 1.8. 

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Vocabulary

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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