Question

problems 1–2: the hangers and the graph represent the same system of equations.
$y = 3x$
$y = x + 3$

1. determine the solution to the system of equations.
2. what does the solution tell you about the weight of a triangle and the weight of a circle that will balance the hanger?

problems 3–4: here is a graph.

1. write an equation that can represent each line.
2. estimate the solution to the system.

Explanation:

Step1: Solve system for Q1

Set equations equal: $3x = x + 3$
Subtract $x$: $2x = 3$
Solve for $x$: $x = \frac{3}{2} = 1.5$
Find $y$: $y = 3(1.5) = 4.5$

Step2: Interpret solution for Q2

Relate $x,y$ to weights: $x$=triangle, $y$=circle

Step3: Find equations for Q3

For red line: slope $m=\frac{0-4}{8-0}=-\frac{1}{2}$, y-intercept $b=4$, so $y=-\frac{1}{2}x+4$
For blue line: slope $m=\frac{2-0}{1-0}=2$, y-intercept $b=0$, so $y=2x$

Step4: Estimate solution for Q4

Find intersection of $y=2x$ and $y=-\frac{1}{2}x+4$:
Set equal: $2x = -\frac{1}{2}x + 4$
Multiply by 2: $4x = -x + 8$
Solve: $5x=8 \implies x=1.6$, $y=2(1.6)=3.2$ (or estimate from graph)

Answer:

1. The solution is $(x,y) = (1.5, 4.5)$
2. A triangle weighing 1.5 lb and a circle weighing 4.5 lb will balance both hangers.
3. Red line: $y = -\frac{1}{2}x + 4$; Blue line: $y = 2x$
4. The estimated solution is $(x,y) = (1.6, 3.2)$ (or approximately $(2, 4)$ from visual estimation of the graph intersection)