SOLUTION: solve by graph: x + y = 3 x + y = -1 2x + y = 4 x + y = 3 solve by addition x + y = 2 x - y = 2 solve by substitution x

SOLUTION: solve by graph: x + y = 3 x + y = -1 2x + y = 4 x + y = 3 solve by addition x + y = 2 x - y = 2 solve by substitution x - y = 12 y = 2x solve by substitution 3x

Question 122666: solve by graph:
x + y = 3
x + y = -1
2x + y = 4
x + y = 3
solve by addition
x + y = 2
x - y = 2
solve by substitution
x - y = 12
y = 2x
solve by substitution
3x - y = -7
x + y = 9
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
I'll do one of each to help you get started:
#1
"solve by graph:
x + y = 3
x + y = -1 "

Start with the given system of equations:

In order to graph these equations, we need to solve for y for each equation.

So let's solve for y on the first equation

Start with the given equation

Subtract from both sides

Rearrange the equation

Divide both sides by

Break up the fraction

Reduce

Now lets graph (note: if you need help with graphing, check out this solver)

Graph of

So let's solve for y on the second equation

Start with the given equation

Subtract from both sides

Rearrange the equation

Divide both sides by

Break up the fraction

Reduce

Now lets add the graph of to our first plot to get:

Graph of (red) and (green)

From the graph, we can see that the two lines are parallel and will never intersect. So there are no solutions and the system is inconsistent.

#2

"solve by addition
x + y = 2
x - y = 2 "

Start with the given system of equations:

Now in order to solve this system by using elimination/addition, we need to solve (or isolate) one variable. I'm going to solve for y.

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for , we would have to eliminate (or vice versa).

So lets eliminate . In order to do that, we need to have both coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.

So to make the coefficients equal in magnitude but opposite in sign, we need to multiply both coefficients by some number to get them to an common number. So if we wanted to get and to some equal number, we could try to get them to the LCM.

Since the LCM of and is , we need to multiply both sides of the top equation by and multiply both sides of the bottom equation by like this:

Multiply the top equation (both sides) by
Multiply the bottom equation (both sides) by

Distribute and multiply

Now add the equations together. In order to add 2 equations, group like terms and combine them

Combine like terms and simplify

Simplify

Divide both sides by to isolate y

Reduce

Now plug this answer into the top equation to solve for x

Start with the first equation

Plug in

Subtract 0 from both sides

Combine like terms on the right side

So our answer is
and
which also looks like

Notice if we graph the equations (if you need help with graphing, check out this solver)

we get

graph of (red) and (green) and the intersection of the lines (blue circle).

and we can see that the two equations intersect at

. This verifies our answer.

#3

"solve by substitution
x - y = 12
y = 2x "

Start with the given system

Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.

Combine like terms on the left side

Divide both sides by -1 to isolate x

Divide

Now that we know that , we can plug this into to find

Substitute for each

Simplify

So our answer is and which also looks like

Notice if we graph the two equations, we can see that their intersection is at

. So this verifies our answer.

Graph of (red) and (green)

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