7.2-3 Solving Linear Equations

A linear equation in one variable is an equation in which the same letter is used in all variable terms and the exponent of the variable is 1.

The solution to an equation in one variable is the number that can be substituted in place of the variable and makes the equation true. For example 5 is a solution to the equation 2x + 3 = 13 because 2(5) + 3 = 13 is true.

Equivalent equations are equations that have the same solutions. For example 2x + 3 = 13 and x = 5 are equivalent equations because each has the solution of 5.

Basic Principle of Equality To preserve equality, if an operation is performed on one side of an equation, the same operation must be performed on the other side.

There are 2 principles (axioms) we will use to solve linear equations in one variable. The first is the addition principle of equality. This principle allows us to add (or subtract) the same value to both sides of an equation to obtain an equivalent equation.

To solve an equation using the addition axiom: Locate the variable in the equation. Identify the constant that is associated with the variable by addition or subtraction. Add the opposite of the constant to both sides of the equation.

Solve each equation.

Often we need to combine like terms on one or both sides of the equation before solving. For example:

Whenever variable terms appear on both sides of an equation we use the addition principle to move all variable terms to the same side, then solve. For example to solve

When solving these type equations, it makes no difference the side from which you remove the variable term to start. The goal is to get all variable terms on one side and all constants on the other.

Solve each equation.

Whenever quantities appear in parentheses on either side of the equation they must be removed first.

Solve each equation.

The Multiplication Principle of Equality

For each problem so far the coefficient for the variable ended up being one. We use the multiplication principle (axiom) to solve equations where the coefficient of the variable is not one.

The multiplication principle allows us to multiply (or divide) each side of an equation by the same nonzero quantity to obtain an equivalent equation. The goal is to get +1 times the variable = a number.

To solve an equation where the coefficient of the variable is not one you need to multiply both sides of the equation by the reciprocal of the coefficient. An alternate way is to divide both sides by the coefficient of the variable.

Solve each equation.

Sometimes it is necessary to combine like terms before solving the equation.

Solve each equation.

When more than one operation is indicated on the variable, undo addition or subtraction first, then undo multiplication or division next.

Solve: 4x – 2 = 18 Since the variable has been multiplied by 4 and subtracted by 2, undo by adding 2 and dividing by 4.

Solve:118 – 22m = 30 Think of 118 – 22m as ( - 22m)

Solve: 5x – 4 = 8x – 13

Summary of steps for solving an equation: Remove parentheses. Combine like terms on each side of the equation. Sort terms to collect the variable terms on one side and constants on the other. Solve for the variable by multiplying by the reciprocal of the coefficient or dividing by the coefficient of the variable.

5 ( h – 4 ) + 2 = 3h – 4 5h – = 3h – 4 Distribute. 5h – 18 = 3h – 4 Step 1 Step 2 Combine terms. 5h – = 3h – Add 18. 5h = 3h + 14Combine terms. Subtract 3h.5h – 3h = 3h + 14 – 3h2 Combine terms.2h = 14 = h = 7 Step 3 2h 14 Divide by 2. Solve the following equation.

Check by substituting 7 for h in the original equation.Step 4 5 ( h – 4 ) + 2 = 3h – 4 5 ( 7 – 4 ) + 2 = 3(7) – 4 5 (3) + 2 = 3(7) – = 21 – 4 17 = 17 ? Let h = 7. ? Subtract. True ? Multiply. The solution to the equation is 7.

Solving an Equation That Has Infinitely Many Solutions 4 ( 2n + 6 ) = 2 ( 3n + 12 ) + 2n 8n + 24 = 6n n Distribute. 8n + 24 = 8n + 24 Combine terms. 8n + 24 – 24 = 8n + 24 – 24 Subtract 24. 8n = 8nCombine terms. Subtract 8n. 8n – 8n = 8n – 8n True0 = 0 An equation with both sides exactly the same, like 0 = 0, is called an identity. An identity is true for all replacements of the variables. We indicate this by writing all real numbers.

Solving an Equation That Has No Solution 6x – ( 4 – 3x ) = ( 3x – 9 ) 6x – 4 + 3x = 8 + 9x – 27Distribute. 9x – 4 = –19 + 9x Combine terms. 9x – 4 – 9x = –19 + 9x – 9xSubtract 9x. – 4 = –19False Again, the variable has disappeared, but this time a false statement (– 4 = – 19) results. Whenever this happens in solving an equation, it is a signal that the equation has no solution and we write no solution. 1