Unit_3_Test_review__1_of_1_v2_ANSWERS

Systems of Equations and Inequalities

1. Intersect Points of Given Systems of Equations

  • Equations:

    • 4x + 3y = -5

    • -2x + 2y = 6

  • Substituting into one equation to solve for y:

    • From -2x + 2y = 6, rearranging gives:

      • 2y = 2x + 6

      • y = x + 3

  • Substituting y in the other equation:

    • 4x + 3(x + 3) = -5

    • 4x + 3x + 9 = -5

    • 7x = -14

    • x = -2

    • Now substitute x back into y = x + 3:

      • y = -2 + 3 = 1

  • Solution Point: (-2, 1)

2. Solutions to Each System of Equations

  • Second System:

    • y = 3x - 2

    • x - y = 4

  • Substitute y in second equation:

    • x - (3x - 2) = 4

    • x - 3x + 2 = 4

    • -2x + 2 = 4

    • -2x = 2

    • x = -1

  • Substitute x back into y = 3(-1) - 2:

    • y = -3 - 2 = -5

  • Solution Point: (-1, -5)

3. Verify Solution Points of the Provided Systems

  • System:

    • 5x + 4y = -7

    • -5x - 2y = 1

  • Checking Points:

    • A. (-1, -3):

      • 5(-1) + 4(-3) = -5 - 12 = -17 (No)

      • -5(-1) - 2(-3) = 5 + 6 = 11 (No)

    • B. (0, 3):

      • 5(0) + 4(3) = 12 (No)

      • -5(0) - 2(3) = -6 (No)

    • C. (1, -3):

      • 5(1) + 4(-3) = 5 - 12 = -7 (Yes)

      • -5(1) - 2(-3) = -5 + 6 = 1 (Yes)

    • D. (-3, 1):

      • 5(-3) + 4(1) = -15 + 4 = -11 (No)

  • Correct Solution Point: C. (1, -3)

4. Solutions to the System of Inequalities

  • Inequalities:

    • -3x + 2y < 1

    • y > -2x + 4

  • Checking Points:

    • A. (3, 1):

      • -3(3) + 2(1) = -9 + 2 = -7 < 1 (Yes)

      • 1 > -2(3) + 4 = -6 + 4 = -2 (Yes)

    • B. (-4, 1):

      • -3(-4) + 2(1) = 12 + 2 = 14 < 1 (No)

    • C. (-1, -3):

      • -3(-1) + 2(-3) = 3 - 6 = -3 < 1 (Yes)

      • -3 > -2(-1) + 4 = 2 + 4 = 6 (No)

    • D. (0, 1):

      • -3(0) + 2(1) = 2 < 1 (No)

  • Correct Solution Point: A. (3, 1)

Additional Problems

5. Solving a Simple Equation

  • If 2x + 3 = 16

  • Solutions:

    • 2x = 16 - 3 = 13

    • 2x - 6 = 13 - 6 = 7

  • x = 6.5: Check; 2(6.5) - 6 = 7

6. And 7. System of Equations for Cost Analysis

  • Betty's Purchases:

    • 2H + 1P = 11

    • 2H + 3P = 17

  • Ramón's Equipment Cost:

    • 6S + 6G = 90

    • 4S + 8G = 100

8. Solution Points from Graphs

  • Solutions Listed:

    • Point (-2, -1)

    • Point (3, -2)

    • No Solution

Equivalence of Equations

9. Checking Equivalence of Two Equations

  • Equations:

    • ⅔ x + 3y = 9

    • 2x + 6y = 27

  • Comparative Method:

    • Multiply first equation by 3 to align x-coefficients:

      • New equation: 2x + 9y = 27

      • Compare to: 2x + 6y = 27

  • Conclusion:

    • Equations are NOT equivalent because resulting forms differ at y-coefficient level.

Graphing Systems of Inequalities

10. Representation of Inequalities

  • Inequality:

    • y > x - 3

  • Y-Intercept: (0, -3)

  • Slope: 1 (upward)

  • Graph Characteristics:

    • Dashed line as it's strictly greater >

    • Shade above the line to represent y values greater than x - 3

Selection of Graph for Inequality Systems

11. Selecting Correct Graph of Inequalites

  • Criteria:

    • A solution to ONE of the inequalities is (1, 5)

    • A solution to ONLY ONE of the inequalities is (-2, 3)

    • A solution to BOTH inequalities is (0, 0)

  • Options:

    • Options (a), (b), (c), (d) to be checked for these conditions.


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