A linear equation is one in which each term is either a constant, a variable to the power of 1 or a combination of both. Examples of linear equations in one variable are:
$2x+3=10$2x+3=10, $5y-2=3\left(2y+5\right)$5y−2=3(2y+5), $\frac{4z+6}{5}=10z+1$4z+65=10z+1
Later in this chapter, non-linear equations will be studied, such as:
$x^2+5x-6=0$x2+5x−6=0 (quadratic) and $2x^3-8x=0$2x3−8x=0 (cubic)
The process of solving linear equations involves trying to find an unknown value. This is achieved by rearranging the equation using inverse operations to isolate the unknown.
While rearranging an equation, it is very important to keep the equation equivalent or balanced. This means that when any mathematical operation is done to one side of the equation, this operation is also done to the other side of the equation. Once a solution is found, this can be checked by substituting the value back into the equation to see if it is true.
In some questions, there may be variables on both sides of the equation. To solve, rearrange the equation so that the terms with the variables are moved to one side of the equation. Sometimes, expansion of the equation may even be required before doing this. Once the terms with variables are moved to one side of the equation, combine like terms and isolate the variable by using inverse operations to find the solution.
Linear equations can be solved by isolating the variable on one side of the equation by using inverse operations.
Practice question
Question 1
Solve the following equation: $2\left(2x+5\right)=3\left(x+5\right)$2(2x+5)=3(x+5)
Working with fractions
When solving equations with fractions, remember to use inverse operations to make the variable the subject of the equation.
It may be helpful to write any fractions as improper fractions rather than mixed numbers. Remove fractions from the equation by multiplying both sides by the lowest common denominator.
Practice questions
Question 2
Solve the following equation:
$\frac{14k-76}{2}-2=2k$14k−762−2=2k
Question 3
Solve $\frac{x+2}{2}+3=\frac{x+3}{7}$x+22+3=x+37 for $x$x.
Problem-solving with equations
The skills required to group variables and solve equations can also be used to solve practical problems in context. First, identify the unknown quantities and represent these with a variable. Next, take note of any other information that relates mathematical operations to the variable. Construct the equation using this information and then solve by applying the same skills for isolating a variable with inverse operations.
Practice question
Question 4
Kate and Isabelle do some fundraising for their sporting team. Together they raised $\$600$$600.
If Isabelle raised $\$p$$p, and Kate raised $\$272$$272 more than Isabelle:
Solve for the value of $p$p.
Write each line of working as an equation.
Now calculate how much Kate raised.