2.04 Solving linear equations
Inverse operations
To solve equations using algebra, the most important rule to remember is that if we apply operations to one side of the equation, we must also apply it to the other.
When applying operations to equations, we always apply the same step to both sides of the equation. This way, both sides of the equation will be equal once we solve the equation.
Making sure to follow this rule, we can isolate the pronumeral in an equation by applying operations to both sides of the equation which reverse the operations applied to the pronumeral.
To do this, we need to know which operations are inverses of each other.
| Operation | Inverse operation | Example |
|---|---|---|
| \text{Addition} \\ \text{Subtraction} | \text{Subtraction} \\ \text{Addition} | x+4-4=x |
| \text{Multiplication} \\ \text{Division} | \text{Division} \\ \text{Multiplication} | y\times4\div4=y |
| \text{Powers} \\ \text{Roots} | \text{Roots} \\ \text{Powers} | \sqrt{x^{2}}=x |
As we are only dealing with linear equations at this stage, powers and roots will not be involved but have been included for reference.
Examples
Example 1
Solve the equation: -x-7=7
When applying operations to equations, we always apply the same step to both sides of the equation. This way, both sides of the equation will be equal once we solve the equation.
Equations with brackets
If we have an equation with one set of brackets such as 3\left(x-5\right)=9 we can either expand the brackets before solving or, in this case as 3 is a factor of 9, divide both sides of the equation by 3. But in cases where we have two sets of brackets, we will first want to expand both sets of brackets before combining like terms. We can then solve the equation by performing inverse operations.
Examples
Example 2
Solve this equation for x: 2\left(2x+5\right) + 3\left(4x+6\right)=76
To solve equations with brackets, expand the brackets using the distributive law, and then use inverse operations to solve for x.
Equations with pronumerals on both sides
To solve equations with variables on both sides of the equation, we want to use inverse operations (usually by adding or subtract terms) to eliminate the variables from one side of the equation. We can then combine like terms and solve using inverse operations.
Examples
Example 3
Solve this equation for x: 5x=x+8
Example 4
Solve this equation for x: \dfrac{3x+8}{7} = \dfrac{-3x+12}{3}
To solve equations with variables on both sides of the equation, we want to use inverse operations to eliminate the variable from one side of the equation. We can then combine like terms and solve using inverse operations.