Question 191937This question is from textbook
: Explain how to
a.Apply elimination in solving a system of equations.
b.Apply substitution in solving a system of equations.
c.Demonstrate each technique in solving the system
2x + 7y = 12
5x – 4y = -13 Thank You!
This question is from textbook
Answer by RAY100(1637)   (Show Source):
You can put this solution on YOUR website! explaining as we go, first elimination
1) 2x+7y=12
2) 5x-4y=-13
elimination means to eliminate one variable (x or y)
we normally do this by adding the two equations, variable side to variable side, constant side to constant side. This does not change equality as we are adding equal values to both sides of an already equal equation.
however, in our case, the coefficients are not similar for either x, or y. Therefore we adjust one or both equations by multiplying both sides by a constant.
In this case we choose to multiply (1) by 4, and (2) by 7 to eliminate the y term
(1) *4 results in 8x+28y=48
(2) *7 results in 35x-28y=-91
adding (1) to (2) results in 43x +0y = -43
dividing both sides by 43
x=(-1)
Typically after we get one variable we substitute that into either equation to solve for the other variable.
in our case let's substitute into (1)
2(-1) +7y=12
-2 +7y =12
add (+2) to both sides
7y=14
divide both sides by 7
y=2
checking both equations by substitution
1) 2(-1) +7(2) = 12 ok
2) 5(-1) -4(2)=-13 ok
Usually elimination can be accomplished by multiplying only one equation if a common multiple can be found. This was a difficult set.
Normally, we like to add the equations to eliminate variables, HOWEVER it is possible to substract the equations to eliminate variables. In this case watch the signs carefully. Both sides of the equation need to be substracted.
In substitution, we pick one equation, solve for a variable, and then substitute that variable in the OTHER equation. Often a coefficient of 1 before a variable facilitates this technique.
In our case, let's start with (1)
2x+7y=12
substract 7y both sides
2x=12-7y
divide both sides by 2
x= (12-7y)/2=6-(7/2)y
substitute this into (2)
5x-4y=-13
5(6-(7/2)y) -4y=-13
distribution 5
30-(35/2)y -4y =-13
subt 30 each side
-(35/2)y -4y = -43
simplifying
-21.5y=-43
divide both sides by (-21.5)
y=2
Again, once we have one variable we find the other by substuting into the base equations and solving for the other variable.
in this case use (1), 2x +7y =12
2x +7(2) = 12
subst 14 each side
2x = -2
divide by 2
x=(-1)
note this is as before.
normally in Linear Equations, Graphing is also included as a solution technique.
In this each equation is graphed and the intersection is the solution (x,y)
Students often have difficulty with the fact that one variable has an assumed number, and the other results from solving the equation. Just use 0 and 1 to get started.
As this form is not extremely accurate, checking the answer is recommended.
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