Question

what is the solution of the system? use a graph
$f(x) = 2x + 4$
$f(x) = x + 2$
$f(x)=2x+4$
slope: $\frac{2}{1}$ y - intercept $4$
$f(x)=x+2$
slope: $\frac{1}{1}$ y - intercept $2$

x	y		x	y	check
-3			-3
-2			-2
-1			-1
0			0
1			1
2			2
3			3

solve the following system
$x + y = 4$
$y = 2x + 1$

x	y		x	y	check
-2			-2
-1			-1
0			0
1			1
2			2
3			3
4			4

Explanation:

First System: $f(x)=2x+4$ and $f(x)=x+2$

Step1: Fill table for $y=2x+4$

Substitute $x$ values into $y=2x+4$:

• $x=-3$: $y=2(-3)+4=-2$
• $x=-2$: $y=2(-2)+4=0$
• $x=-1$: $y=2(-1)+4=2$
• $x=0$: $y=2(0)+4=4$
• $x=1$: $y=2(1)+4=6$
• $x=2$: $y=2(2)+4=8$
• $x=3$: $y=2(3)+4=10$

Step2: Fill table for $y=x+2$

Substitute $x$ values into $y=x+2$:

• $x=-3$: $y=-3+2=-1$
• $x=-2$: $y=-2+2=0$
• $x=-1$: $y=-1+2=1$
• $x=0$: $y=0+2=2$
• $x=1$: $y=1+2=3$
• $x=2$: $y=2+2=4$
• $x=3$: $y=3+2=5$

Step3: Find intersection point

Set $2x+4=x+2$, solve for $x$:
$2x - x = 2 - 4$
$x=-2$
Substitute $x=-2$ into $y=x+2$: $y=0$

Step1: Rewrite $x+y=4$ as $y=4-x$

Rearrange to solve for $y$: $y=4-x$

Step2: Fill table for $y=4-x$

Substitute $x$ values into $y=4-x$:

• $x=-2$: $y=4-(-2)=6$
• $x=-1$: $y=4-(-1)=5$
• $x=0$: $y=4-0=4$
• $x=1$: $y=4-1=3$
• $x=2$: $y=4-2=2$
• $x=3$: $y=4-3=1$
• $x=4$: $y=4-4=0$

Step3: Fill table for $y=2x+1$

Substitute $x$ values into $y=2x+1$:

• $x=-2$: $y=2(-2)+1=-3$
• $x=-1$: $y=2(-1)+1=-1$
• $x=0$: $y=2(0)+1=1$
• $x=1$: $y=2(1)+1=3$
• $x=2$: $y=2(2)+1=5$
• $x=3$: $y=2(3)+1=7$
• $x=4$: $y=2(4)+1=9$

Step4: Find intersection point

Set $4-x=2x+1$, solve for $x$:
$4-1=2x+x$
$3=3x$
$x=1$
Substitute $x=1$ into $y=4-x$: $y=3$

Answer:

Table for $y=2x+4$:

X	Y
-2	0
-1	2
0	4
1	6
2	8
3	10

Table for $y=x+2$:

X	Y
-2	0
-1	1
0	2
1	3
2	4
3	5

Solution of the system: $(-2, 0)$

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Second System: $x+y=4$ and $y=2x+1$