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Solve Linear Equations

Solve Linear Equations. Section 2.1 MATH 116-460 Mr. Keltner. Some definitions. An equation is a mathematical statement that two expressions are equal. A linear equation with one variable is an equation that can be written in the form ax + b = 0,

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Solve Linear Equations

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  1. Solve Linear Equations Section 2.1 MATH 116-460 Mr. Keltner

  2. Some definitions • An equation is a mathematical statement that two expressions are equal. • A linear equation with one variable is an equation that can be written in the form ax + b = 0, where a and b are constants and a ≠ 0.

  3. More definitions • A number is a solution of an equation if substituting the number in place of the variable produces a true statement. • Ex.: x = 3 is a solution of the equation 2x -7 = -1. Just get to the assignment! • Two equations are equivalent equations if they have the same solution(s). • An example of this might be 2x=4 and 8x = 16.

  4. Inverse Operations • The big goal in solving equations is to isolate the variable, or get the variable by itself. • There is a list of properties on pg. 57 in your book of how to solve equations. • I prefer to think of inverse operations, where you try to undo different parts of the equation. • Addition and subtraction undo one another. • Multiplication and division undo one another.

  5. Example 1A variable in one place • Solve 2/3x - 7 = 5.

  6. Example 2Make your own equation • Your total cost for buying a DVD, including 7% sales tax, is $23.54. What was the price of the DVD before sales tax was added?

  7. Example 3A variable on both sides • What is the solution of the equation 8y - 16 = 13y + 9?

  8. Example 4Using the distributive property • Solve 4(2x - 9) + 5x = -3(10 - x).

  9. This LCD is pretty sweet, but can it make fractions disappear? • Instead of having to work with fractions in an equation, you can multiply the equation by the least common denominator (or LCD). • This eliminates the fractions altogether!

  10. Example 5Bye, bye, fractions! • 3/7w - 2/9 = 4/9w + 1/7

  11. But what about decimals? • There is also a shortcut for eliminating decimals from equations. • It deals with identifying the longest decimal in the entire equation. • Just multiply the entire equation (both sides) by a 1, followed by as many zeros as there are decimal places in the longest decimal. • The equation 2.25b + 3.8 = 1.75b + 5.2 would be multiplied by 100, since its longest decimal is 2 places.

  12. Example 6Later, decimals! 2.25b + 3.81 = 1.75b + 5.26

  13. Contradictions We say there is “No Solution” to an equation when we simplify to a false statement with no variables. Example: 4(x + 3) = 4x + 7 Identities We say that “All real numbers” are solutions to an equation when we simplify to a true statement with no variables. Example: 3x + 9 = 1.5 (2x + 6) Special cases “No solution” and “All Real Numbers”

  14. Formulas: Solving for a Specific Variable • When solving for a specific variable, you will still use inverse operations. • Work to gather all terms that have the specified variable on one side of the equation. • Isolate the specific variable just like you would when solving with constants.

  15. Example 7: Solving a Formula • If a fundraising project produces a profit of P dollars from a revenue of R dollars, while having a cost of C dollars, it is described by the formula P = R - C • Solve this formula for C.

  16. Example 8: Try one more formula • Solve the formula below for F. C = 5/9(F - 32)

  17. Assessment Pgs. 64-65: #’s 9 - 42 and 51 - 60, multiples of 3

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