Question

solve each system of equation using the graphing method

1. solve each system.

a. $y = - 2x + 5$
$y=\frac{1}{8}x-\frac{7}{2}$
b. $3x - y = 5$
$4x - 2y = 4$
c. $y = - 2x + 3$
$y = x$
d. $2x + y = 5$
$4x - 1 = y$

Explanation:

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Part a

Step1: Set equations equal

Since $y=-2x+5$ and $y=\frac{1}{8}x-\frac{7}{2}$, set them equal:
$$-2x+5=\frac{1}{8}x-\frac{7}{2}$$

Step2: Eliminate fractions

Multiply all terms by 8:
$$-16x+40=x-28$$

Step3: Isolate x terms

$$-16x - x = -28 - 40$$
$$-17x=-68$$

Step4: Solve for x

$$x=\frac{-68}{-17}=4$$

Step5: Solve for y

Substitute $x=4$ into $y=-2x+5$:
$$y=-2(4)+5=-8+5=-3$$

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Part b

Step1: Rewrite in slope-intercept form

For $3x-y=5$: $y=3x-5$
For $4x-2y=4$: Divide by 2: $2x-y=2$, so $y=2x-2$

Step2: Set equations equal

$$3x-5=2x-2$$

Step3: Solve for x

$$3x-2x=-2+5$$
$$x=3$$

Step4: Solve for y

Substitute $x=3$ into $y=3x-5$:
$$y=3(3)-5=9-5=4$$

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Part c

Step1: Set equations equal

Since $y=-2x+3$ and $y=x$:
$$x=-2x+3$$

Step2: Isolate x terms

$$x+2x=3$$
$$3x=3$$

Step3: Solve for x

$$x=1$$

Step4: Solve for y

Since $y=x$, $y=1$

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Part d

Step1: Rewrite first equation

For $2x+y=5$: $y=5-2x$

Step2: Set equations equal

Since $y=4x-1$, set equal to $y=5-2x$:
$$4x-1=5-2x$$

Step3: Isolate x terms

$$4x+2x=5+1$$
$$6x=6$$

Step4: Solve for x

$$x=1$$

Step5: Solve for y

Substitute $x=1$ into $y=4x-1$:
$$y=4(1)-1=3$$

Answer:

a. $(4, -3)$
b. $(3, 4)$
c. $(1, 1)$
d. $(1, 3)$