SOLUTION: I am really having a hard time understanding solving using substitution 2x+3y=24 -3x-5y=15 how do I solve those using substitution

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Question 1190343: I am really having a hard time understanding solving using substitution
2x+3y=24
-3x-5y=15
how do I solve those using substitution
Found 3 solutions by Theo, MathLover1, greenestamps:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
with substitution, you solve for one of the variable in terms of the other variable in one of the equations and then you use that value to solve for the other variable in the other equation.

it's a way of reducing two unknown variable to one unknown variable.

a simpler example would be like this.

consider these two equations.

x + y = 1
2x + 3y = 10

solve for x in the first equation to get:
x = 1 - y

replace x in the second equation to get:
2 * (1 - y) + 3y = 10
simplify to get:
2 - 2y + 3y = 10
combine like terms to get:
2 + y = 10
solve for y to get:
y = 10 - 2 = 8.

now that you know the value of y, solve for x in either equation to get:

x + y = 1 becomes x + 8 = 1 which becomes x = -7.

your values are x = -7 and y = 8

go back to the original equations and replace a with -7 and y with 8.
you will get:

x + y = 1 becomes -7 + 8 = 1 which becomes 1 = 1 which is true.
2x + 3y = 10 becomes -14 + 24 = 10 which becomes 10 = 10 which is true.
this confirms the values of x and y are good because the same value of x and y make both equations true.

that was just an example.

your problem is a little more complicated because of the arithmetic involved but the concept is the same.

you are solving for one of the variables in terms of the other variable in one of the equation and you are using the result of that to solve for the other variable in the other equations.

your equations are:

2x + 3y = 24
-3x - 5y = 15.

take the first equation and solve for y in terms of x.

you will get x = (24 - 3y) / 2.

replace x with that in the second equation.

you will get:

-3x - 5y = 15 becomes:
-3 * [(24 - 3y) / 2] - 5y = 15
multiply both sides of that equation by 2 to get:
-3 * (24 - 3y) - 10y = 30
simplify to get:
-3 * 24 + 3 * 3y - 10y = 30
simplify further to get:
-72 + 9y - 10y = 30
combine like terms to get:
-72 - y = 30
add 72 to both sides of the equation to get:
-y = 102
solve for y to get:
y = -102

now that you know the value of y, you can use that value of y to solve for x in either of the original two equations.

2x + 3y = 24 becomes 2x + 3 * -102) = 24 which becomes 2x - 306 = 24 which becomes 2x = 330 which becomes x = 165.

-3x - 5y = 15 becomes -3x - 5 * -102 = 15 which becomes -3x + 510 = 15 which becomes -3x = -495.
solve for x to get:
x = -495 / -2 = 165.

your solution is that x = 165 and y = -102.

replace x and y with those values in the original equations and you will get:

2x+3y=24 becomes 2*165 - 3*102 = 24 which becomes 24 = 24 2 which is true.
-3x-5y=15 becomes -3*165 +5*102 = 15 which becomes 15 = 15 which is also true.
this confirm the solution is correct.

here's a reference on solving a system of equations simultaneously with substitution.

https://www.purplemath.com/modules/systlin4.htm











Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

.....eq.1
....eq.2
_____________________________

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations




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

Solve for y for the first equation

Subtract from both sides

Divide both sides by 3.


Which breaks down and reduces to



Now we've fully isolated y

Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x.


Replace y with . Since this eliminates y, we can now solve for x.

Distribute -5 to

Multiply



Reduce any fractions

Add to both sides


Combine the terms on the right side



Make -3 into a fraction with a denominator of 3

Now combine the terms on the left side.


Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get



<---------------------------------One answer

Now that we know that , lets substitute that in for x to solve for y

Plug in into the 2nd equation

Multiply

Add to both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out -5 on the left side.

Multiply the terms on the right side


Reduce


So this is the other answer


<---------------------------------Other answer


So our solution is

and

which can also look like

(,)

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




we get


graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


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


-----------------------------------------------------------------------------------------------
Check:

Plug in (,) into the system of equations


Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies



Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies


Since the solution (,) satisfies the system of equations






this verifies our answer.
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!



(1) I personally would never solve a pair of equations like this using substitution. When the equations are both in this form ("Ax+By=C"), I would always use elimination.

2x+3y=24
-3x-5y=15

multiply the first equation by 3 and the second by 2 so that the x terms are 6x and -6x; then adding the two equations eliminates x. You can then solve for y, and once you have y you can use either original equation to solve for x.

6x+9y=72
-6x-10y=30
-y=102
y=-102

2x+3(-102)=24
2x-306=24
2x=330
x=165

ANSWER: x=165; y=-102


(2) Solving by substitution, you solve one equation for one variable in terms of the other and substitute that into the other equation.

solve the first equation for x:
2x+3y=24
2x=-3y+24
x=-1.5y+12

substitute that expression for x in the second equation:
-3(-1.5)y+12)-5y=15
4.5y-36-5y=15
-.5y=51
y=51/-.5=-102

x=(-1.5)(-102)+12=153+12=165

Same answer; but the work is a lot uglier....
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