Question

1. shanna drew a map of elmwood on a coordinate plane. there are two bike paths in the town.

a. graph and label each equation on the coordinate plane.
b. estimate the solution of the system and explain what it represents.

1. critique reasoning brianna was asked to use graphing to solve the system of equations ( y = 2x + 3.5 ) and ( y = 2.5x - 1 ).

a. brianna said the lines are parallel and do not intersect, so the system has no solution. do you agree or disagree? explain.
b. another student said the solution is ( (9, 21.5) ). is he correct? explain.

Explanation:

For Question 4B:

Step1: Identify equations

The two equations are $y = -3x - 2$ (Riverside Path) and $y = -\frac{1}{2}x + 1$ (Hilltop Path).

Step2: Find intersection visually

From the graph, the lines cross at $x \approx -1.2$, $y \approx 1.6$.

Step3: Verify with substitution

Set $-3x - 2 = -\frac{1}{2}x + 1$
$-3x + \frac{1}{2}x = 1 + 2$
$-\frac{5}{2}x = 3$
$x = -\frac{6}{5} = -1.2$
Substitute $x=-1.2$ into $y = -\frac{1}{2}x + 1$:
$y = -\frac{1}{2}(-1.2) + 1 = 0.6 + 1 = 1.6$

Parallel lines have identical slopes. The slope of $y=2x+3.5$ is 2, and the slope of $y=2.5x-1$ is 2.5. Since the slopes are different, the lines are not parallel and will intersect.

Step1: Substitute $x=9$ into first equation

$y = 2(9) + 3.5 = 18 + 3.5 = 21.5$

Step2: Substitute $x=9$ into second equation

$y = 2.5(9) - 1 = 22.5 - 1 = 21.5$

Answer:

The solution is $(-1.2, 1.6)$. This represents the coordinate point where the two bike paths intersect on the town map.

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For Question 5A: