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Complete the tables below. Note the inclusion of decimal and fractional values as per senior phase requirements. Using the skill of algebraic substitution, calculate the output values for the following linear equations.

Mathematics

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Answer

1.5\textbf{1.5}

3\textbf{3}

Step-by-step solution

6 steps

Task 1.1: Complete the tables

Equation 1: y=12x+2y = \frac{1}{2}x + 2

  1. Calculate output (y) values for Equation 1.

    • For x=−2x = -2: y=12(−2)+2=−1+2=1y = \frac{1}{2}(-2) + 2 = -1 + 2 = 1
    • For x=−1x = -1: y=12(−1)+2=−0.5+2=1.5y = \frac{1}{2}(-1) + 2 = -0.5 + 2 = 1.5
    • For x=0x = 0: y=12(0)+2=0+2=2y = \frac{1}{2}(0) + 2 = 0 + 2 = 2
    • For x=0.5x = 0.5: y=12(0.5)+2=0.25+2=2.25y = \frac{1}{2}(0.5) + 2 = 0.25 + 2 = 2.25
    • For x=2x = 2: y=12(2)+2=1+2=3y = \frac{1}{2}(2) + 2 = 1 + 2 = 3
  2. Complete the table for Equation 1.

    Input (x)−2−100.52Output (y)11.522.253Ordered Pair (x; y)(−2;1)(−1;1.5)(0;2)(0.5;2.25)(2;3)\begin{array}{|c|c|c|c|c|c|} \hline \text{Input (x)} & -2 & -1 & 0 & 0.5 & 2 \\ \hline \text{Output (y)} & 1 & 1.5 & 2 & 2.25 & 3 \\ \hline \text{Ordered Pair (x; y)} & (-2; 1) & (-1; 1.5) & (0; 2) & (0.5; 2.25) & (2; 3) \\ \hline \end{array}

    Show your substitution calculations for Equation 1 when:

  3. Calculate for x=−1x = -1.

    y=12x+2y=12(−1)+2y=−12+2y=−0.5+2y=1.5\begin{align*} y &= \frac{1}{2}x + 2 \\ y &= \frac{1}{2}(-1) + 2 \\ y &= -\frac{1}{2} + 2 \\ y &= -0.5 + 2 \\ y &= \boxed{\textbf{1.5}} \end{align*}
  4. Calculate for x=2x = 2.

    y=12x+2y=12(2)+2y=1+2y=3\begin{align*} y &= \frac{1}{2}x + 2 \\ y &= \frac{1}{2}(2) + 2 \\ y &= 1 + 2 \\ y &= \boxed{\textbf{3}} \end{align*}

    Equation 2: y=2x−4y = 2x - 4

  5. Calculate output (y) values for Equation 2.

    • For x=−2x = -2: y=2(−2)−4=−4−4=−8y = 2(-2) - 4 = -4 - 4 = -8
    • For x=−1x = -1: y=2(−1)−4=−2−4=−6y = 2(-1) - 4 = -2 - 4 = -6
    • For x=0x = 0: y=2(0)−4=0−4=−4y = 2(0) - 4 = 0 - 4 = -4
    • For x=0.5x = 0.5: y=2(0.5)−4=1−4=−3y = 2(0.5) - 4 = 1 - 4 = -3
    • For x=2x = 2: y=2(2)−4=4−4=0y = 2(2) - 4 = 4 - 4 = 0
  6. Complete the table for Equation 2.

    Input (x)−2−100.52Output (y)−8−6−4−30Ordered Pair (x; y)(−2;−8)(−1;−6)(0;−4)(0.5;−3)(2;0)\begin{array}{|c|c|c|c|c|c|} \hline \text{Input (x)} & -2 & -1 & 0 & 0.5 & 2 \\ \hline \text{Output (y)} & -8 & -6 & -4 & -3 & 0 \\ \hline \text{Ordered Pair (x; y)} & (-2; -8) & (-1; -6) & (0; -4) & (0.5; -3) & (2; 0) \\ \hline \end{array}
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