Solution of an Equation
A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.
Determine whether a number is a solution to an equation.
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
In the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.
EXAMPLE
Determine whether
y=43 is a solution for
4y+3=8y.
Solution:
|
4y+3=8y |
Substitute 43 for y |
4(43)+3=?8(43) |
Multiply. |
3+3=?6 |
Add. |
6=6✓ |
Since
y=43 results in a true equation,
43 is a solution to the equation
4y+3=8y.
Now it is your turn to determine whether a fraction is the solution to an equation.
We introduced the Subtraction and Addition Properties of Equality in Solving Equations Using the Subtraction and Addition Properties of Equality. In that section, we modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.
EXAMPLE
Solve:
x+11=−3.
Answer:
Solution:
To isolate x, we undo the addition of 11 by using the Subtraction Property of Equality.
|
x+11=−3 |
Subtract 11 from each side to "undo" the addition. |
x+11−11=−3−11 |
Simplify. |
x=−14 |
Check: |
x+11=−3 |
|
Substitute x=−14 . |
−14+11=?−3 |
|
|
−3=−3✓ |
|
Since
x=−14 makes
x+11=−3 a true statement, we know that it is a solution to the equation.
Now you can try solving an equation that requires using the addition property.
In the original equation in the previous example,