SOLUTION: On the below 4 equations, can you show me the steps either by substitution or addition method to determine the 2 digit graph intersection points on a graph? Problem 1: x+2y =

Algebra ->  Graphs -> SOLUTION: On the below 4 equations, can you show me the steps either by substitution or addition method to determine the 2 digit graph intersection points on a graph? Problem 1: x+2y =      Log On


   

Question 1171000: On the below 4 equations, can you show me the steps either by substitution or addition method to determine the 2 digit graph intersection points on a graph?
Problem 1:
x+2y = 15
x-2y = -9
Problem 2:
x-y = 0
7x-3y = 24
Problem 3:
3x-8y = 10
x-4y = 3
Problem 4:
2x+3y = -5
3x-y = 20
Thank you
Found 2 solutions by MathLover1, Alan3354:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Problem 1:
x%2B2y+=+15
x-2y+=+-9

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B2%2Ay=15
1%2Ax-2%2Ay=-9

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

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

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 1 is 1, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -1 like this:

1%2A%281%2Ax%2B2%2Ay%29=%2815%29%2A1 Multiply the top equation (both sides) by 1
-1%2A%281%2Ax-2%2Ay%29=%28-9%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
1%2Ax%2B2%2Ay=15
-1%2Ax%2B2%2Ay=9

Notice how 1 and -1 add to zero (ie 1%2B-1=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%281%2Ax-1%2Ax%29%2B%282%2Ay%2B2%2Ay%29=15%2B9

%281-1%29%2Ax%2B%282%2B2%29y=15%2B9

cross%281%2B-1%29%2Ax%2B%282%2B2%29%2Ay=15%2B9 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

4%2Ay=24

y=24%2F4 Divide both sides by 4 to solve for y



y=6 Reduce


Now plug this answer into the top equation 1%2Ax%2B2%2Ay=15 to solve for x

1%2Ax%2B2%286%29=15 Plug in y=6


1%2Ax%2B12=15 Multiply



1%2Ax=15-12 Subtract 12 from both sides

1%2Ax=3 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%283%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=6

which also looks like

(3, 6)

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

1%2Ax%2B2%2Ay=15
1%2Ax-2%2Ay=-9

we get



graph of 1%2Ax%2B2%2Ay=15 (red) 1%2Ax-2%2Ay=-9 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (3,6). This verifies our answer.



Problem 2:
x-y+=+0
7x-3y+=+24
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax-1%2Ay=0
7%2Ax-3%2Ay=24

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

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

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 7 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 7 is 7, we need to multiply both sides of the top equation by 7 and multiply both sides of the bottom equation by -1 like this:

7%2A%281%2Ax-1%2Ay%29=%280%29%2A7 Multiply the top equation (both sides) by 7
-1%2A%287%2Ax-3%2Ay%29=%2824%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
7%2Ax-7%2Ay=0
-7%2Ax%2B3%2Ay=-24

Notice how 7 and -7 add to zero (ie 7%2B-7=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%287%2Ax-7%2Ax%29-7%2Ay%2B3%2Ay%29=0-24

%287-7%29%2Ax-7%2B3%29y=0-24

cross%287%2B-7%29%2Ax%2B%28-7%2B3%29%2Ay=0-24 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-4%2Ay=-24

y=-24%2F-4 Divide both sides by -4 to solve for y



y=6 Reduce


Now plug this answer into the top equation 1%2Ax-1%2Ay=0 to solve for x

1%2Ax-1%286%29=0 Plug in y=6


1%2Ax-6=0 Multiply



1%2Ax=0%2B6 Subtract -6 from both sides

1%2Ax=6 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%286%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=6 Multiply the terms on the right side


So our answer is

x=6, y=6

which also looks like

(6, 6)

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

1%2Ax-1%2Ay=0
7%2Ax-3%2Ay=24

we get



graph of 1%2Ax-1%2Ay=0 (red) 7%2Ax-3%2Ay=24 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (6,6). This verifies our answer.



Problem 3:
3x-8y+=+10
x-4y+=+3
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax-8%2Ay=10
1%2Ax-4%2Ay=3

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

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

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 3 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 1 is 3, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -3 like this:

1%2A%283%2Ax-8%2Ay%29=%2810%29%2A1 Multiply the top equation (both sides) by 1
-3%2A%281%2Ax-4%2Ay%29=%283%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
3%2Ax-8%2Ay=10
-3%2Ax%2B12%2Ay=-9

Notice how 3 and -3 add to zero (ie 3%2B-3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%283%2Ax-3%2Ax%29-8%2Ay%2B12%2Ay%29=10-9

%283-3%29%2Ax-8%2B12%29y=10-9

cross%283%2B-3%29%2Ax%2B%28-8%2B12%29%2Ay=10-9 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

4%2Ay=1

y=1%2F4 Divide both sides by 4 to solve for y



y=1%2F4 Reduce


Now plug this answer into the top equation 3%2Ax-8%2Ay=10 to solve for x

3%2Ax-8%281%2F4%29=10 Plug in y=1%2F4


3%2Ax-8%2F4=10 Multiply



3%2Ax-2=10 Reduce



3%2Ax=10%2B2 Subtract -2 from both sides

3%2Ax=12 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%2812%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=4 Multiply the terms on the right side


So our answer is

x=4, y=1%2F4

which also looks like

(4, 1%2F4)

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

3%2Ax-8%2Ay=10
1%2Ax-4%2Ay=3

we get



graph of 3%2Ax-8%2Ay=10 (red) 1%2Ax-4%2Ay=3 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (4,1%2F4). This verifies our answer.




Problem 4:
2x%2B3y+=+-5
3x-y+=+20
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

2%2Ax%2B3%2Ay=-5
3%2Ax-1%2Ay=20

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

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

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 2 and 3 to some equal number, we could try to get them to the LCM.

Since the LCM of 2 and 3 is 6, we need to multiply both sides of the top equation by 3 and multiply both sides of the bottom equation by -2 like this:

3%2A%282%2Ax%2B3%2Ay%29=%28-5%29%2A3 Multiply the top equation (both sides) by 3
-2%2A%283%2Ax-1%2Ay%29=%2820%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
6%2Ax%2B9%2Ay=-15
-6%2Ax%2B2%2Ay=-40

Notice how 6 and -6 add to zero (ie 6%2B-6=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%286%2Ax-6%2Ax%29%2B%289%2Ay%2B2%2Ay%29=-15-40

%286-6%29%2Ax%2B%289%2B2%29y=-15-40

cross%286%2B-6%29%2Ax%2B%289%2B2%29%2Ay=-15-40 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

11%2Ay=-55

y=-55%2F11 Divide both sides by 11 to solve for y



y=-5 Reduce


Now plug this answer into the top equation 2%2Ax%2B3%2Ay=-5 to solve for x

2%2Ax%2B3%28-5%29=-5 Plug in y=-5


2%2Ax-15=-5 Multiply



2%2Ax=-5%2B15 Subtract -15 from both sides

2%2Ax=10 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ax=%2810%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=5 Multiply the terms on the right side


So our answer is

x=5, y=-5

which also looks like

(5, -5)

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

2%2Ax%2B3%2Ay=-5
3%2Ax-1%2Ay=20

we get



graph of 2%2Ax%2B3%2Ay=-5 (red) 3%2Ax-1%2Ay=20 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (5,-5). This verifies our answer.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
On the below 4 equations, can you show me the steps either by substitution or addition method to determine the 2 digit graph intersection points on a graph?
Problem 1:
x+2y = 15
x-2y = -9
--------------------- Add
2x = 24
x = 12
Sub for x and find y.
===================================
Problem 2:
x-y = 0
7x-3y = 24
y = x
Sub for either.
==========================
Problem 3:
3x-8y = 10
x-4y = 3
Multiply the 2nd eqn by 2, then subtract.
===============================================
Problem 4:
2x+3y = -5
3x-y = 20
Multiply the 2nd eqn by 3 then add.