Algebra

Linear Equations

Big Picture

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.

Key Terms

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.

Linear Equations in One Variable

A linear equation can include:

  • Constant (number that does not contain any variables)
  • A constant times a single variable raised to the first power only (\(x^{1}\))

Example: \(x + 5 = 2x + 3\) is a linear equation

  • x, 5, 2x, and 3 are all terms
  • x and 2x are both a constant times a variable raised to the first power
  • 5 and 3 are constants
  • 2 is the coefficient of x
  • x and 2x are also like terms - they both only contain the x variable

Example: \(2xy + 3 = x^{2}\) is an equation but not a linear equation

  • \(x^{2}\) is not raised to the first power
  • 2xy is not a constant times a single variable

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.

  • A variable in an equation represents a number that makes the equation true
  • To solve an equation is to find the value of the variable that makes the equation true

Properties of Equality

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.

  • Addition Property of Equality: If \(a = b\), then \(a + c = b + c\)
  • Subtraction Property of Equality: If \(a = b\), then \(a - c = b - c\)
  • Multiplication Property of Equality: If \(a = b\), then \(ac = bc\)
  • Division Property of Equality: If \(a = b\) and \(c ≠ 0\), then \(\frac{a}{c} - \frac{b}{c}\)
  • Reflexive Property of Equality: \(a = a\)
  • Symmetric Property of Equality: If \(a = b\), then \(b = a\)
  • Transitive Property of Equality: If \(a = b\) and \(b = c\), then \(a = c\)

Algebra

Linear Equations cont.

Solving Equations

Using Addition/Subtraction

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.

Example:

\(x - 3 = 2\)
\(x - 3 + 3 = 2 + 3\)   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.

Example:

\(x + 4 = 20\)
\(x + 4 - 4 = 20 - 4\)   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.

Using Multiplication/Division

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.

  • Multiplying by the reciprocal is the same as dividing by the original number.

Example:

\(4x = 24\)
\(4x . \frac{1}{4} = 24 . \frac{1}{4}\)    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

Using Addition/Subtraction

Ratio can be written in three ways: a to \(b, a:b, \frac{a}{b}\)

  • Should be written in simplest form
  • Example: 1 apple:3 oranges (1:3 ratio)

Proportions are used to solve problems

  • Cross-multiply method is used to solve proportions
  • The numerator of each ratio is multiplied by the denominator of the other ratio

Example:

\(\frac{x}{6} = \frac{1}{3}\)
\(3x = 6\)
\(x = 2\)

This is the same as using the multiplication property of equality.

\(\frac{x}{6} . 6 . 3 = \frac{1}{3} . 6 . 3\)
\(x . 3 = 6\)
\(x = 2\)

Algebra

Number Lines cont.

Multi-Step Equations

The steps below are some guidelines for solving multi-step equations. Make sure each step obeys the properties of equality and order of operations.

  • Simplify as much as you can using order of operations (PEMDAS) and the distributive property.
  • Isolate (move) the variable to one side by using inverse operations.
  • Divide both sides of the equation by the variable’s coefficient (as long as the coefficient  is not 0).

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.

Equations with Variables on Both Sides

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

Number of Solutions

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.

  • Examples: \(2 = 3, x + 1 = x +3\)

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.

  • Examples: \(3 = 3, x = x\)

Notes