Solving Linear Equations

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Presentation transcript:

Solving Linear Equations 1.5 through 1.7

Equations Is 2 + 4 = 6 a true equation? Is 3 – 5 = 10 a true equation? Is 2x + 1 = 7 a true equation? This equation is a conditional equation, since it depends on x.

Solving Linear Equations: The graph of any linear equation is a straight line. Solving Linear Equations: If there are fractions, multiply everything by the LCD. Get rid of ( )’s and combine like terms. Isolate the variable (get it by itself on one side of equation). Check your answer!

Example 1 Determine whether 3 is a solution of 2x + 4 = 10 Solution To find out, check the proposed solution, x = 3: 2x + 4 = 10 This is the original equation. 2(3) + 4 = 10 Substitute 3 for x. 6 + 4 = 10 Multiply. 10 = 10 Add. Since 10 = 10, 3 is a solution, or root, of the equation.

Example 3 Solve 3(x - 2) = 20. 3(x - 2) = 20 3x – 6 = 20 3x = 26 x = Solution: 3(x - 2) = 20 Original equation. 3x – 6 = 20 Use the distributive property. 3x = 26 Add 6 to both sides. x = Divide both sides by 3.

Example 5 Solve 10(x - 3) = 9(x - 2) + 12 10x - 30 = 9x - 18 + 12 Solution: This is the given equation. Multiply both sides by 6 (LCD). Use distributive property. 10(x - 3) = 9(x - 2) + 12 10x - 30 = 9x - 18 + 12 Use distributive property again. 10x – 30 = 9x - 6 Combine like terms. x = 24 Isolate variable.

Solving Linear Equations: The graph of any linear equation is a straight line. Solving Linear Equations: If there are fractions, multiply everything by the LCD. Get rid of ( )’s and combine like terms. Isolate the variable (get it by itself on one side of equation). Check your answer!

Types of Equations Conditional: An equation that is true for at least one value of “x” (it’s sometimes true). Identity: An equation that is true for all values of “x” (it’s always true). Contradiction: An equation that has no solution (it’s never true).

Example 7 Solve 2(x +1) – x = 3(1 + x) – (2x + 1): Original equation. 2x + 2 – x = 3 + 3x – 2x – 1 Use the distributive property. x + 2 = x + 2 Combine like terms. Last equation is always true for any value of x it is an identity!

Example 8 Solve Since –2 = 5 is never true, there is no solution Original equation. Multiply everything by 6 (LCD). Simplify fractions. Use distributive property. 26x – 2 = 26x + 5 Combine like terms. -2 = 5 Subtract 26x from both sides. Since –2 = 5 is never true, there is no solution it is a contradiction!

Time Trial This equation is a contradiction. Determine whether the equation 3(x - 1) = 2(x + 3) + x is an identity, a conditional equation, or a contradiction. Solution To find out, solve the equation. 3(x – 1) = 2(x + 3) + x 3x – 3 = 2x + 6 + x 3x – 3 = 3x + 6 -3 ≠ 6 This equation is a contradiction.

Formulas A formula is an equation involving two or more variables. D = RT A = LW P = 2L + 2W

Example 10 Solve for t in the following formula: A = p + prt. Solution: To solve for t means to isolate it on on side. A = p + prt Original equation. A – p = prt To isolate t, subtract p from both sides. Divide both sides by pr. Write t on left-hand side.