An equation is a mathematical statement where two expressions are equal to each other. An equation is true only if the expressions have the same value. For example, the numerical equation \( 15 = 15 \) is true. While this example contains only numbers, equations can have numbers, operations, and variables. When an equation contains a variable, it is called an algebraic equation, and you can usually use inverse operations to solve it. A number is a solution to an equation only when its substitution back into the equation creates a true statement.
For Example
Is \( p = 2.4 \) the solution to \( 1.5p = 3.6 \)? Use substitution to decide.
The steps to solve this problem are in the table below. Read the step in the left-hand column, and try to apply it to the example problem on your own. Then click the step when you are ready to check your work.
Replace the variable \( p \) with the number 2.4 in the equation. \( \displaystyle 1.5(2.4) = 3.6 \) |
|
Remember that the parentheses indicate multiplication. \( \displaystyle 1.5(2.4) = 3.6 \) \( 3.6 = 3.6 \) |
|
If the result is a true statement, then the number is a solution. The statement \( 3.6 = 3.6 \) is true. \( p = 2.4 \) is a solution. If the result had been a false statement, the number would NOT have been a solution. |
Let's Practice
Practice determining if a given number is a solution to an equation with the activity below. Read the question on the front of each box. Determine if the value is a solution of the equation shown. You can click the box to check your answer.
Is \( p = 4 \) a solution for \( p - 7 = 11 \)?
No. Partial work is shown:
\( \displaystyle (4) - 7 = 11 \)
\( \displaystyle 4 + \left( - 7 \right) = 11 \)
\( - 3 \neq 11 \)
Is \( y = 6 \) a solution for \( - 3y = - 18 \)?
Yes. Partial work is shown:
\( \displaystyle - 3(6) = - 18 \)
\( - 18 = - 18 \)
Is \( g = - 2 \) a solution for \( 4 - g = 6 \)?
Yes. Partial work is shown:
\( \displaystyle 4 - \left( - 2 \right) = 6 \)
\( \displaystyle 4 + 2 = 6 \)
\( 6 = 6 \)
Is \( x = 15 \) a solution for \( \frac{x}{5} = - 3 \)?
No. Partial work is shown:
\( \displaystyle \frac{(15)}{5} = - 3 \)
\( \displaystyle 15 \div 5 = - 3 \)
\( 3 \neq - 3 \)
