Question 1116108: Given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solution. Show why this is true by solving the system of equations given. Justify the reason for each step. HINT: Use addition property of equality and multiplication property of equality in your answers.
5x + 2y = 7
3x – y = 2
A1. To solve the system using elimination, first multiply the bottom equation by 2. Write the new system of equations.
B1. Why is this multiplication allowed?
C1. What variable will be eliminated when the equations are combined after the multiplication?
D1. Next, add the equations together. Your answer should be a single equation with one variable.
E1. Why can you add the equations?
F1. Solve the equation for x.
G1. Substitute x back into one of the equations to solve for y.
H1. What is the solution to the system of equations?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 5x + 2y = 7
3x – y = 2
multiply the second equation by 2 and leave the first equation as is to get:
5x + 2y = 7
6x – 2y = 4
add the second equation to the first equation to get:
11x = 11
divide both sides of that equation by 11 to get:
x = 1
replace x with 11 in the first original equation to get:
5x + 2y = 7 becomes 5 + 2y = 7
solve for y to get y = 1
your solution is x = 1 and y = 1
confirm by replacing x and y with 1 and 1 in both original equations to get:
5x + 2y = 7 becomes 5 + 2 = 7 which is true.
3x – y = 2 becomes 3 - 1 = 2 which is true.
both original equations are true when x = 1 and y = 1, so that's your solution.
why can you get away with doing this?
it's because of the following properties of equalities.
if a = b, then a + c = b + c
you're adding the same thing to both sides of the equation, so the equality still holds.
consider a = 5
if a = b, then b must be equal to 5.
add 2 to both sides of the equation.
you get 5 + 2 = 5 + 2 which results in 7 = 7 which is true.
if a = b, then ac = bc
you're multiplying both sides of the equation by the same thing, so the equality still holds.
consider a = 5.
then b must be equal to 5, because a = b
if c = 2, then 2*5 = 2*5 becomes 10 = 10 which is true.
in your equation, these principles are put into practice.
you multiplied both sides of the second equation by 2.
the equality still holds true.
note that the value of each side of the equation has changed, but the equality still holds.
now what happens when you add both equations together, and why can you go this.
you have 5x + 2y = 7 added to 6x - 2y = 4
when you add equations together, you have to combine like terms and you added the left side of the equation and the right side of the equation separately.
so you get 5x + 6x = 11x and you get 2y - 2y = 0 and you get 7 + 4 = 11, with the result being 11x + 0y = 11 which becomes 11x = 11 which allows you to solve for x to get x = 1
now if you let a = 3x + 2y and you let b = 7, then your first equation becomes a = b
now if you let 6x = 2y = c and you let 4 = d, then your second equation becomes c = d.
when you add both equations together, you get a = b + c = d becomes a + c = b + d.
but since c = d, then you can replace d with c and the equation of a + c = b + d becomes a + c = b + c and you already know that, if you add the same quantity to both sides of an equation, the equality remains the same.
so, it all comes back to the properties of equality.
here's a list of the properties of equality from https://www.varsitytutors.com/hotmath/hotmath_help/topics/properties-of-equality
here's some good tips on how to solve equations.
https://www.mathsisfun.com/algebra/equations-solving.html
here's some god tips on how to solve systems of equations.
https://www.mathsisfun.com/algebra/systems-linear-equations.html
now that that's all done, let's see how you might go about answering the questions asked.
A1. To solve the system using elimination, first multiply the bottom equation by 2. Write the new system of equations.
B1. Why is this multiplication allowed?
because if you multiply both sides of an equation by the same value, then the equality is preserved and the equation is still true.
C1. What variable will be eliminated when the equations are combined after the multiplication?
you would eliminate y because 2y - 2y = 0.
D1. Next, add the equations together. Your answer should be a single equation with one variable.
E1. Why can you add the equations?
you can add the two equations together, because you are adding the same value to each side of the equation, since the left side of an equation is equal to the right side of the same equation which makes them equivalent in value to each other.
F1. Solve the equation for x.
x = 1
G1. Substitute x back into one of the equations to solve for y.
y = 1
H1. What is the solution to the system of equations?
x = 1 and y = 1
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