Question

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graph each system of equations and identify t

1. $\begin{cases} y = -3x + 4 \\ y = 4x - 3 end{cases}$

solution:

Explanation:

Step1: Find intersection algebraically

Set equations equal:
$$-3x + 4 = 4x - 3$$
$$4 + 3 = 4x + 3x$$
$$7 = 7x$$
$$x = 1$$
Substitute $x=1$ into $y=-3x+4$:
$$y = -3(1) + 4 = 1$$

Step2: Identify line 1 intercepts

For $y=-3x+4$:

• y-intercept: $x=0$, $y=4$ → $(0, 4)$
• x-intercept: $y=0$, $0=-3x+4$ → $x=\frac{4}{3} \approx 1.33$ → $(\frac{4}{3}, 0)$

Step3: Identify line 2 intercepts

For $y=4x-3$:

• y-intercept: $x=0$, $y=-3$ → $(0, -3)$
• x-intercept: $y=0$, $0=4x-3$ → $x=\frac{3}{4}=0.75$ → $(\frac{3}{4}, 0)$

Step4: Plot and verify graph

Plot intercepts for each line, draw lines; they cross at $(1,1)$.

Answer:

The solution to the system is the intersection point $(1, 1)$.
To graph:

1. For $y=-3x+4$: plot $(0,4)$ and $(\frac{4}{3},0)$, draw the line.
2. For $y=4x-3$: plot $(0,-3)$ and $(\frac{3}{4},0)$, draw the line.
3. The lines intersect at $(1, 1)$.