Algebra Essentials

D The number of hours the plumber worked

QUESTION 8 Systems of Linear Equations

Find the coordinate intersection solution \((x, y)\) for the following system of linear equations: \(y = 2x + 1\)  and  \(y = -x + 10\)

A \((3, 7)\)

B \((4, 9)\)

C \((2, 5)\)

D \((9, 1)\)

ALGEBRA ESSENTIALS PRACTICE ASSIGNMENT PAGE 2 OF 2

SOLUTION 6 Correct Answer: B

Identify valid mathematical function

Definition: A function cannot map the same input (\(x\)-value) to different outputs (\(y\)-values). In coordinates, no \(x\)-value may repeat with a different \(y\).

• Option A repeats input \(1\): \((1, 5)\) and \((1, 7)\). (NOT a function)
• Option B: \(\{(3, 2), (4, 2), (5, 3), (6, 3)\}\) has unique \(x\)-values. (VALID FUNCTION)
• Option C repeats input \(5\) four times. (NOT a function)
• Option D repeats input \(0\): \((0, 4)\) and \((0, -4)\). (NOT a function)

SOLUTION 7 Correct Answer: B

Meaning of the constant value \(85\) in the equation \(C = 45h + 85\)

In any linear relationship \(y = mx + b\):

• The coefficient \(m\) (\(45\)) represents the variable rate (labor rate of \$45.00/hour).
• The constant \(b\) (\(85\)) represents the starting point when hours \(h = 0\). This represents a one-time flat house call fee of \$85.00 before any work starts.

SOLUTION 8 Correct Answer: A

Solve the system: \(y = 2x + 1\) and \(y = -x + 10\)

Step 1: Set equations equal to each other (Substitution Method):
    \(2x + 1 = -x + 10\)
Step 2: Solve for \(x\) by adding \(x\) and subtracting \(1\):
    \(3x = 9 \rightarrow x = 3\)
Step 3: Plug \(x = 3\) back in to find \(y\):
    \(y = 2(3) + 1 \rightarrow y = 7\). Intersection coordinate is \(\mathbf{(3, 7)}\).

ALGEBRA ESSENTIALS ANSWER KEY PAGE 2 OF 2