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Math: Basic Tutorials : Multiple-Step Linear Equations
Multiple-Step Linear Equations
Multiple-step linear equations are equations that involve multiple operations (e.g. subtraction and multiplication). This section explains how to solve these equations using the order of operations.
Examples
Click on the titles below to view each example.
Solve 6n minus 2 equals negative 3n plus 7, and verify the solution.
Line 1: We want to remove the variables from the right side of the equation, so we add 3n to both sides to leave only constants on the right. The equation becomes 6n plus 3n minus 2 equals negative 3n plus 3n plus 7.
Line 2: Combine the like terms to simplify so the equation becomes 9n minus 2 equals 7.
Line 3: We want to remove the constants from the left side of the equation, so we add 2 to both side and the equation becomes 9n minus 2 plus 2 equals 7 plus 2.
Line 4: Simplify to get 9n equals 7.
Line 5: To isolate for n we must undo the multiplication, so we divide both sides by 9 and the equation becomes 9n over 9 equals 9 over 9.
Line 6: Simplify to get the solution n equals 1.
Line 7: Check you answer in the given equation 6n minus 2 equals negative 3n plus 7.
Line 8: Substitute x equals 1 into the equation, so it is 6 times 1 minus 2 equals negative 3 times 1 plus 7.
Line 9: Complete the multiplication to simplify the equation to 6 minus 2 equals negative 3plus 7.
Line 10: Simplify to get 4 equals 4.
Since n equals 1 makes 6n minus 2 equals negative 3n plus 7 a true statement, 1 is the solution to the equation.
Solve 3 times open bracket x minus 2 close bracket minus 5 equals 4 times open bracket 2x plus 1 close bracket plus 5, and verify the solution.
Line 1: Distribute to remove the brackets so the equation becomes 3x minus 6 minus 5 equals 8x plus 4 plus 5.
Line 2: Combine like terms to simplify so the equation becomes 3x minus 11 equals 8x plus 9.
Line 3: Since the numerical coefficient 8 is greater than 3 we will subtract 3x from both sides of the equation, so it becomes 3x minus 3x minus 11 equals 8x minus 3x plus 9.
Line 4: Simplify to get the equation negative 11 equals 5x plus 9.
Line 5: Subtract 9 from both sides of the equation to get the constants on the left, so the equation is negative 11 minus 9 equals 5x plus 9 minus 9.
Line 6: Simplify to get negative 20 equals 5x.
Line 7: Divide both sides by 5 to isolate x so the solution to the equation is negative 4 equals x.
Line 8: Check you answer in the given equation 3 times open bracket x minus 2 close bracket minus 5 equals 4 times open bracket 2x plus 1 close bracket plus 5.
Line 9: Substitute x equals negative 4 into the equation, so it is 3 times open bracket negative 4 minus 2 close bracket minus 5 equals 4 times open bracket 2 time negative 4 plus 1 close bracket plus 5.
Line 10: Simplify inside the brackets so the equation is 3 times negative 6 minus 5 equals 4 time negative 7 plus 5.Line 11: Multiply to simplify the equation to negative 18 minus 5 equals negative 28 plus 5.
Line 12: Simplify to get negative 23 equals negative 23.
Since x equals negative 4 makes 3 times open bracket x minus 2 close bracket minus 5 equals 4 times open bracket 2x plus 1 close bracket plus 5 a true statement, negative 4 is the solution to the equation.
Summary and Worksheet
- Summary: Solving Multi-Step Linear Equations - PDF - Opens in a new windowThis document contains a short (1 – 2 page) summary of this topic as well as detailed examples to illustrate key concepts. Use this summary to review this topic.
- Worksheet: Multiple-Step Linear Equations - PDF - Opens in a new windowThis document contains practice questions on this topic. Use the worksheet to test your knowledge and practice the skills learned in this module. The answers to the practice questions are provided at the end.
Attribution
Examples ource: "Prealgebra - opens in a new window" by Lynn Marecek & Mary Anne Anthony-Smith is licensed under CC BY 4.0 - opens in a new window / A derivative from the original work - opens in a new window