Question

practice it
pick-3 practice: number of solutions
choose any three of the problems below and complete them.

1. given the graph of ( y = 3x - 9 ) and ( y = -\frac{1}{2}x + 12 )

a. identify the number of solutions
b. if possible, find one solution to the system. if there are no solutions, justify how you know.

1. you are given the graph of ( y = -\frac{1}{3}x - 6 ) and ( y = \frac{1}{5}x + 10 )

how many solutions are there to this system? explain how you know.

1. how many solutions are there to the system of equations ( y = \frac{2}{3}x - 7 )

and ( y = \frac{2}{3}x + 2 )? justify your response.

1. given the system of equations: ( y = 3x + 4 ) and ( 2y = 6x + 8 )

a. how many solutions are there? explain your reasoning.
b. if possible, find one solution to the system. if there are no solutions, justify how you know.

1. write a system of equations with zero solutions. graph the system.
2. given the equation ( y = -\frac{1}{2}x + 5 ), use this desmos graph to

write a second equation that creates a system with a solution at ( (4, 3) ).

Explanation:

I will solve problems 1, 3, and 4 as the chosen three problems.

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Problem 1

Step1: Count intersections from graph

The two lines cross at one point.

Step2: Find intersection coordinates

Set equations equal: $3x - 9 = -\frac{1}{2}x + 12$
Multiply by 2: $6x - 18 = -x + 24$
Solve for $x$: $7x = 42 \implies x=6$
Substitute $x=6$: $y=3(6)-9=9$

Step1: Compare slopes of the lines

First line slope: $\frac{2}{3}$, second line slope: $\frac{2}{3}$

Step2: Compare y-intercepts

First line y-intercept: $-7$, second line y-intercept: $2$
Parallel lines (same slope, different intercepts) never intersect.

Step1: Simplify the second equation

Divide $2y=6x+8$ by 2: $y=3x+4$

Step2: Compare to first equation

The simplified second equation is identical to $y=3x+4$.

Step3: Identify a solution

Any point on $y=3x+4$ works; substitute $x=0$: $y=3(0)+4=4$

Answer:

a. 1 solution
b. $(6, 9)$

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Problem 3