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Linear Equations - Understanding and Solving

Learn to solve linear equations involving fractions, understand x and y intercepts, and master functions in this comprehensive lesson. Practice problems included.

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Linear Equations - Understanding and Solving

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  1. Warm-up Determine the x-intercept (s) and y-intercept from the graph below. Determine what this viewing rectangle illustrates. 2. [-20, 40, 5] by [-10, 30, 2] 3. Solve this equation: 4x + 5 = 29 • Homework: • pg. 104, (1-45 odds) • Problems 10-14 must show checking your answer.

  2. Answers: • 1. x-intercepts = (3,0) and (-7,0) y-intercept = (0,21) it’s a reflection • 2. The x window’s min is at -20, max at 40 increasing in increments of 5. The y window’s min is at -10, max at 30 increasing in increments of 2. • 3. x = 6

  3. Announcements: Ch 1 Learning Goal: The student will be able to understand functions by solving and graphing all types of equations and inequalities. Today’s Objective: Be able to solve a rational equations with variables

  4. Lesson 1.2A Linear Equations and Rational Equations A linear equation in one variable x is an equation that can be written in the form of ax + b = 0, where a and b are real numbers and a 0.

  5. Steps for Solving a Linear Equation: 1. Simplify the algebraic expression on each side by removing grouping symbols and combining like terms. 2. Collect all the variable terms on one side and all the numbers, or constant terms, on the other side. 3. Isolate the variable and solve. 4. Check the proposed solution in the original equation.

  6. Example 1: Solving a Linear Equation Involving Fractions Given. Find LCD. Solve by Distributive Property and combine like terms. 3(x+2)-4(x-1)=24 3x +6-4x+4 = 24 -x = 14 Solve and check. x = -14

  7. You try these: 4(2x +1) = 29 + 3(2x -5) Answers: x = 5, x=1

  8. Example 2: 10 = 2x +15 x = -5/2 You Try: Answer: x = 3

  9. Example 3: Solving Rational Equations Write problem. Objective – try to clear fractions. Multiply both sides by (x-3) to cancel it out on the left side Distribute the (x-3) on the right side Cross out the (x-3) to get rid of them in denominators

  10. Example Continued X = 3 + 9(x-3) X = 3 + 9x – 27 X = 9x – 24 -8x = -24 X = 3 Simplify Distribute the 9 Subtract 3 and -27 Subtract 9x from both sides Divide by -8. Unfortunately not a solution because of the excluded values. The solution is an empty set, 0 .

  11. You Try! Write problem Multiply both sides by 3(x-2) Distribute the 3(x-2) to the right side Cancel out any 3(x-2) Multiply

  12. Distribute the negative Solve for x. Not a solution because of the excluded values. The solution is an empty set, 0

  13. Summary: What is a rational equation? Give an example of this type of equation. Homework: WS1.1 pg. 104, (1-45 odd) Problems 10-14 must show checking your answer.

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