- Find the value of a variable that satisfies an equation
- Write equations for lines using different forms: slope-intercept, point-slope, and standard form
- Recognize and write equations for horizontal and vertical lines
- Determine if lines are parallel or perpendicular, and write equations for lines parallel or perpendicular to a given line
Solving Linear Equations in One Variable
A linear equation is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form [latex]ax + b = 0[/latex] and are solved using basic algebraic operations.
An equation will always contain an equal sign with an expression on each side. Think of an equal sign as meaning “the same as.” Some examples of equations are [latex]y = mx + b[/latex], [latex]\Large\frac{3}{4}\normalsize x = v + {3} - r[/latex], and [latex]2(6-d) + f(3+k) = \Large\frac{1}{4}\normalsize d[/latex].
We can classify linear equations in one variable as one of three types:
- Identity equation: True for all values of the variable.
- Example: [latex]3x = 2x + x[/latex] The solution set is all real numbers.
- Conditional equation: True for only some values of the variable.
- Example: [latex]5x + 2 = 3x - 6[/latex] Solution: [latex]x = -4[/latex]
- Inconsistent equation: Results in a false statement.
- Example: [latex]5x - 15 = 5(x - 4)[/latex] This leads to [latex]-15 \neq -20[/latex], which is a false statement.
The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[/latex], the variable is [latex]x[/latex], a coefficient is [latex]10[/latex], a term is [latex]10x[/latex], and an expression is [latex]2x-3^2[/latex].
Solving linear equations involves the fundamental properties of equality and basic algebraic operations. Some equations can be solved quickly in your head. For example, what is the value of [latex]y[/latex] in the equation [latex]2y=6[/latex]? You can easily determine that [latex]y=3[/latex] by dividing both sides by [latex]2[/latex].
Other equations are more complicated. Solving [latex]\displaystyle 4\left(\frac{1}{3}\normalsize t+\frac{1}{2}\normalsize\right)=6[/latex] without writing anything down is difficult! That is because this equation contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation, one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.
Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The addition property of equality and the multiplication property of equality explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you will keep both sides of the equation equal.
- (Optional) Multiply to clear any fractions or decimals.
- Simplify each side by clearing parentheses and combining like terms.
- Add or subtract to isolate the variable term—you may have to move a term with the variable.
- Multiply or divide to isolate the variable.
- Check the solution.
Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[/latex].
To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It does not matter which term gets moved, [latex]4x[/latex] or [latex]2x[/latex]; however, to avoid negative coefficients, you can move the smaller term.