Question

1.4 solving systems graphically form a
name cyndy valentine
date __________ period __________

1. graph and solve the system.

y = 1/2x - 3
y=-x + 3
(4, -1)

1. graph and solve the system.

3x + y = 5
x - 2y = 4

1. without graphing, determine whether the system is independent, dependent, or inconsistent. explain what this means graphically.

4x + y = 8
x + 1/4y = 6

Explanation:

1.

Step1: Set the two equations equal

Set $\frac{1}{2}x - 3=-x + 3$.

Step2: Add $x$ to both sides

$\frac{1}{2}x+x-3=-x+x + 3$, which simplifies to $\frac{3}{2}x-3 = 3$.

Step3: Add 3 to both sides

$\frac{3}{2}x-3 + 3=3 + 3$, getting $\frac{3}{2}x=6$.

Step4: Solve for $x$

Multiply both sides by $\frac{2}{3}$, so $x = 6\times\frac{2}{3}=4$.

Step5: Find $y$ - value

Substitute $x = 4$ into $y=-x + 3$, then $y=-4 + 3=-1$.

Step1: Rewrite the first - equation in slope - intercept form

From $3x + y=5$, we get $y=-3x + 5$.

Step2: Rewrite the second - equation in slope - intercept form

From $x-2y = 4$, we have $-2y=-x + 4$, then $y=\frac{1}{2}x-2$.

Step3: Set the two equations equal

$-3x + 5=\frac{1}{2}x-2$.

Step4: Add $3x$ to both sides

$-3x+3x + 5=\frac{1}{2}x+3x-2$, which simplifies to $5=\frac{7}{2}x-2$.

Step5: Add 2 to both sides

$5 + 2=\frac{7}{2}x-2 + 2$, getting $7=\frac{7}{2}x$.

Step6: Solve for $x$

Multiply both sides by $\frac{2}{7}$, so $x = 2$.

Step7: Find $y$ - value

Substitute $x = 2$ into $y=-3x + 5$, then $y=-3\times2+5=-1$.

Step1: Rewrite the equations in slope - intercept form

For $4x + y=8$, we have $y=-4x + 8$.
For $x+\frac{1}{4}y=6$, multiply through by 4 to get $4x + y=24$, then $y=-4x+24$.

Step2: Analyze the slopes and y - intercepts

The slopes of both lines are $m=-4$. The y - intercepts are $b_1 = 8$ and $b_2=24$.
Since the slopes are equal and the y - intercepts are different, the lines are parallel.
In a system of linear equations, parallel lines mean the system is inconsistent. Graphically, inconsistent systems have no solution because the lines never intersect.

Answer:

$(4,-1)$

2.