A mathematical expression is like a “phrase” that includes numbers, operations, and variables, while a mathematical equation is like a “sentence” where two expressions are set equal to each other. Solving linear equations with one variable involves using properties of equality and the order of operations to solve for that unknown variable.
Constant: A quantity that doesn’t change.
Variable: Symbol used to represent a value, quantity, or process.
Coefficient: The constant value that a variable is multiplied by.
Term: The part of an expression or equation that is not an operator. Could be a constant, a variable, or a constant times a variable.
Like Terms: Terms that contain the same variable or combination of variables.
Inverse Operations: Operations that undo each other. Addition and subtraction are inverse operations, and so are multiplication and division.
Ratio: A way to compare two numbers, measurements, or quantities.
Proportion: A way to compare two ratios or make equivalent fractions.
A linear equation can include:
Example: \(x + 5 = 2x + 3\) is a linear equation
Example: \(2xy + 3 = x^{2}\) is an equation but not a linear equation
The goal in solving an equation is to isolate the variable by itself on one side of the equals sign. The variable will then be equal to the value on the other side of the equals sign.
Properties of equality are useful for understanding how to rearrange and solve equations. When solving equations, both sides of the equations must be equal at every step.
Equations are solved by using inverse operations. To solve an equation that involves subtraction, add the subtracted number to both sides to isolate the variable.
\(x - 3 = 2\)
\(x - 3 + 3 = 2 + 3\) Add 3 to both sides
\(x = 5\)
Check: \(5 - 3 = 2\)? Yes - answer is correct
To solve an equation that involves addition, subtract the added number from both sides of the equation to isolate the variable.
\(x + 4 = 20\)
\(x + 4 - 4 = 20 - 4\) Substract 4 from both sides
\( x = 16\)
Check: \(16 + 4 = 20\)? Yes - answer is correct
Checking your answers is a quick way to find out if you made a mistake. Plug your answer back into the equation by replacing the variable with the value of the solution. If your answer is right, then the equation should stay true.
To solve an equation that involves multiplication or division, multiply both sides by the reciprocal of the variable’s coefficient in order to isolate the variable.
\(4x = 24\)
\(4x . \frac{1}{4} = 24 . \frac{1}{4}\) Multiply each side by the reciprocal of 4 to isolate x.
\(x = 6\) The reciprocal of 4 is \(\frac{1}{4}\)
This is the same as using the multiplication property of equality.
Check: \(4 • 6 = 24\)? yes - answer is correct
Ratio can be written in three ways: a to \(b, a:b, \frac{a}{b}\)
Proportions are used to solve problems
\(\frac{x}{6} = \frac{1}{3}\)
\(3x = 6\)
\(x = 2\)
This is the same as using the multiplication property of equality.
\(\frac{x}{6} . 6 . 3 = \frac{1}{3} . 6 . 3\)
\(x . 3 = 6\)
\(x = 2\)
The steps below are some guidelines for solving multi-step equations. Make sure each step obeys the properties of equality and order of operations.
It is usually easier to not work with fractions, so try to remove them whenever you can. Multiply both sides of the equation by the lowest common multiple of the denominators.
You can solve equations with variables on both sides just like you solve multi-step equations. To isolate the variable on one side, combine like terms.
Example of like terms: \(3x + 2x = (3 + 2)x = 5x\)
You can then divide both sides of the equation by the variable’s coefficient.
Example: Solve for x given \(- \frac{2}{3}(1 + 3x) = 4x + 6\)
\(3 . - \frac{2}{3}(1 + 3x) = 3 . (4x + 6) \)
Remove fractions by multiplying both sides by 3.
\(-2(1 + 3x) = 12x + 18\)
Distributive property
\(-2 - 6x = 12x + 18\)
\(-2 - 6x + 6x = 12x + 6x + 18\)
Combine like terms by adding 6x to both sides of the equation
\(-2 - 18 = 18x + 18 - 18\)
Isolate the variable by subtracting 18 from both sides.
\(-20 . \frac{1}{18} = 18x . \frac{1}{18}\)
Multiply both sides by the reciprocal of 18
\(x = -\frac{20}{18}\)
Symmetric property of equality
\(x = -\frac{10}{9} \)
Simplify fractions
Usually there is only one value for the variable that will make the equation true. There are two other special cases:
If at any point the two sides of the equation are no longer equal to each other, the equation is false and has no solution. This means no value for the variable will ever make the equation true.
Check your work to make sure you didn’t make a mistake that causes the two sides of the equation to be no longer equal.
If the equation simplifies to an identity, all numbers will make the equation true, and the equation has an infinite number of solutions.