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Solving Linear Equations

Learn how to solve multi-step equations by isolating variables through carefully structured mathematical operations. This comprehensive guide covers principles, examples, and essential techniques for solving complex equations efficiently.

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Solving Linear Equations

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  1. Solving Linear Equations Multi-Step Equations

  2. Solving Multi-Step Equations • To find the solution to an equation we must isolate the variable. • We isolate the variable by performing operations that will eliminate (cancel) the other numbers from the expression.

  3. Solving Multi-Step Equations • We have seen how to eliminate a constant (Addition Property of Equality) & how to eliminate a coefficient (Multiplication Property of Equality). What if an equation has both a constant and a coefficient to be eliminated? • This is called a Multi-Step Equation.

  4. Solving Multi-Step Equations • This is a multi-step equation. 3x + 5 = 17 • We must eliminate the constant, 5, and the coefficient, 3, to isolate the variable, x. • The order that we use the equality properties is very important, we must follow the order of operations in reverse!

  5. Solving Multi-Step Equations • In the expression 3x + 5, the order of operations tells us to multiply 3 times x, and then add the product to 5. Eliminating in reverse order, we need to eliminate the 5 first by adding (-5): 3x + 5 + (-5) = 17 + (-5) 3x + 0 = 12 3x = 12.

  6. Solving Multi-Step Equations • Now we need to eliminate the 3 by multiplying 1/3:

  7. Replace x with 4. Multiply. Add. 17 = 17. √ Solving Multi-Step Equations • Check the solution: 3x + 5 = 17 3(4) + 5 = 17 12 + 5 = 17

  8. Solving Multi-Step Equations • The Steps: • If the variable is on the right side of the equation, swap sides. • If there is a constant on the variable side, add the opposite of the constant to both sides of the equation. • If there is a coefficient in front of the variable, multiply both sides of the equation by the reciprocal of the coefficient.

  9. Change subtraction to addition. (Keep-Change-Change.) Add (+4) to both sides. Multiply both sides by 1/3. Examples Check: 3k - 4 = 8 3(4) - 4 = 8 12 - 4 = 8 8 = 8 √

  10. Add (-1) to both sides. Multiply both sides by 1/5. Examples Check: 5x + 1 = -14 5(-3) +1 = -14 -15 + 1 = -14 -14 = -14 √

  11. Add (-1) to both sides. Multiply both sides by (-1/2). Examples Check: -2x + 1 = 1 -2(0) +1 = 1 0 + 1 = 1 1 = 1 √

  12. Try These!

  13. Solutions!

  14. Solutions!

  15. Solutions!

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