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Solving Linear Equations

Solving Linear Equations. Using Addition/Subtraction. Linear Equation. An equation is an open sentence that joins two expressions with an equal sign. An equation is linear if the variable is to the first power. Examples: 3x + 8 = 9 2b = 4b - 8 3(m + 8) = 2 - 4(m + 1) + 2m.

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Solving Linear Equations

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  1. Solving Linear Equations Using Addition/Subtraction

  2. Linear Equation • An equation is an open sentence that joins two expressions with an equal sign. • An equation is linear if the variable is to the first power. • Examples: • 3x + 8 = 9 • 2b = 4b - 8 • 3(m + 8) = 2 - 4(m + 1) + 2m

  3. Solving Linear Equations • Solving an equation refers to finding the one (or more) values that makes the equation true. • For example, in the equation: x - 3 = 5the solution is 8 because when we substitute 8 for the variable, the equation is true. 8 - 3 = 5

  4. Solving 1-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. • The Addition Property of Equality says that we can add the same number to both sides of an equation without changing the solution to the equation.

  5. Addition Property of Equality • Take the equation: x - 3 = 5 • When we see subtraction, we will make a habit of rewriting the expression (Keep-Change-Change.): x + (-3) = 5 • Our goal is to add a number to both sides of the equation that will isolate the x by canceling the (-3).

  6. Addition Property of Equality • What number when added to (-3) will give us zero? (-3) + 3 = 0 • The Addition Property of Equality says we can add 3 to the equation as long as we add it to both sides. x + (-3) + 3 = 5 + 3

  7. The Identity Property of Addition says that x + 0 = x. Addition Property of Equality • See what happens: x + (-3) + 3 = 5 + 3 x + 0 = 8 x = 8 • The SOLUTION is x = 8. The value 8 makes the final and original equations true.

  8. Checking the Solution • Always check the solution by substituting the value back into the original equation. x - 3 = 5 8 - 3 = 5 5 = 5 • The sentence should be true.

  9. Solving by Addition/Subtraction • Rewrite subtraction as addition (Keep-Change-Change.). • Add the value to both sides that will cancel with the constant (isolating the variable). • Opposite numbers cancel; if the constant is negative, add a positive number; if the constant is positive, add a negative number. • Never add constants that are on different sides of the equal sign!!

  10. Examples 1) x + 9 = 4 x + 9 + (-9) = 4 + (-9) x + 0 = -5 x = -5 Check: (-5) + 9 = 4 4 = 4 √

  11. Examples 2) k - 13 = -5 k + (-13) = -5 k + (-13) + (13) = -5 + (13) k + 0 = 8 k = 8 Check: (8) + (-13) = -5 -5 = -5 √

  12. The Commutative Property of Addition says that w + 4 = 4 + w. Examples 3) -4 + w = 20 -4 + w + (4) = 20 + (4) -4 + (4) + w = 20 + (4) 0 + w = 24 w = 24 Check: -4 +(24) = 20 20 = 20 √

  13. The Symmetric Property of Equality says that; if 8 = 3 + r, then 3 + r = 8. Examples 4) 8 = 3 + r 3 + r = 8 3 + r + (-3) = 8 + (-3) 0 + r = 5 r = 5 Check: 8 = 3 +(5) 8 = 8√

  14. Try These! 1) y - 7 = -8 2) 3 + a = -4 3) 2 = x - (-7) 4) 0 = 9 + r

  15. Solutions! 1) y - 7 = -8 y + (-7) = -8 y + (-7) + (7) = -8 + (7) y + 0 = -1 y = -1 2) 3 + a = -4 3 + a + (-3) = -4 + (-3) 0 + a = -7 a = -7

  16. Solutions! 3) 2 = x - (-7) x - (-7) = 2 x + 7 = 2 x + 7 + (-7) = 2 + (-7) x + 0 = -5 x = -5 4) 0 = 9 + r 9 + r = 0 9 + r + (-9) = 0 + (-9) 0 + r = -9 r = -9

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