Ideas
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.
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.
Examples
Example 1
Solve the equation: -x-7=7
When applying operations to equations, we always apply the same operation to both sides of the equation. This way, both sides of the equation will be equal once we solve the equation.
To solve an equation we apply inverse operations to isolate the variable.
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 need to first expand both sets of brackets before collecting like terms. We can then solve the equation by performing inverse operations.
Examples
Example 2
Solve the following equation: 2\left(3x-5\right)+3\left(4x+6\right)=62
For an equation with one set of brackets, we can either expand the brackets first or divide both sides of the equation by the term outside the brackets.
For equations with two sets of brackets, we first expand both sets of brackets before collecting like terms.
We can then solve the equation by performing inverse operations.
Equations with fractions
Similarly, if we have an equation with a fraction such as \dfrac{2x}{5}=4, we can just perform inverse operations to solve, but for an equation involving the addition or subtraction of two, or more, fractions we multiply both sides of the equation by the lowest common multiple of the denominators to eliminate the fractions. We can then proceed as normal to solve.
Examples
Example 3
Solve the following equation: \dfrac{2x}{4}-\dfrac{2x}{3}=-5
For an equation involving the addition or subtraction of two, or more, fractions we multiply both sides of the equation by the lowest common multiple of the denominators to eliminate the fractions.
Equations with pronumerals on both sides
To solve equations with pronumerals on both sides of the equation, we want to move all pronumerals to one side of the equation. We can then collect like terms and solve using inverse operations.
Examples
Example 4
Solve the following equation: 3x+6=3\left(5x-4\right)+42
To solve equations with pronumerals on both sides of the equation, we move all pronumerals to one side of the equation. We can then collect like terms and solve using inverse operations.