In this lesson we will learn how to apply The Properties of Equality to Systems of Equations so that we can manipulate and solve them.
Additive Property - if a = b is true, a + c = b + c is also true
Multiplicative Property - if a = b is true, ac = bc is also true
What we will learn in this lesson is how we can apply this method simultaneously to multiple equations in a Systems of Equations. Generally, we can think of the format as this:
If, a, b, c, d, e, f are all numbers, and x and y are our variables all in a true System of equations that looks like:
ax + by = c
dx + ey = f
We can multiply one equation or both by any number, say g, and it still remains true, for example
(dx + ey = f ) g
The system remains true.
We can multiply both equations in the system by two different numbers, say g and h,
(ax + by = c) h
(dx + ey = f) g
(dx + ey = f) g - remembering we must multiply both sides of the equation by g
(dx + ey) g = (f ) g
We can multiply both equations in the system by two different numbers, say g and h.
The same is true for the Additive Property.
We can add any number to one equation or both, say g, and it still remain true, for example:
(dx + ey) + g = (f) + g
The system still remains true.
We can add any numbers to both equations in the system by two different numbers, say g and h
(ax + by) + h = (c) + h
(dx + ey) + g =(f) + g
We can also apply the additive property to add or subtract the equations in the system from each other, as long as we equate both sides,
ax + by = c
For example, if we take the first equation and subtract the second equation, the system will still remain true, as we are staying consistent with the “equals sign”.
- (dx + ey = f)
ax - (dx) + by - (ey) = c - (f)
The equation will still remain true, as we are following the properties of equality.
Let’s get into some examples.
The solutions in the above animation are repeated with a more detailed explanation below.
This is is a multi-part question: We have the point (5, 1) as a solution to the system of x + 2y = 7 and 2y - 2x = -8
Firstly, add both equations together, then see if (5, 1) still remains a solution.
Following our additive property, we add the equations together.
x + 2y = 7
+(2y - 2x = -8)
x + (-2x) + 2y + (2y) = 7 + (-8)
Equating like terms gives us,
-x + 4y = -1
Now we sub in (5, 1),
-(5) + 4(1) = -1
-5 + 4 = -1
-1 = -1
As we can see, after adding these equations together, the solution still remains true, as we followed the correct properties of equality.
Next, multiply the first equation (x + 2y = 7) by 2, then check if (5, 1) is still a solution.
Multiply x + 2y = 7 by 2, remembering to multiply both sides
2(x + 2y) = 2(7)
2x + 4y = 14
Now sub in (5, 1)
2(5) + 4(1) = 14
10 + 4 = 14
14 = 14
(5, 1) is still a solution.
Finally, with the equation created after multiplying by 2, add the second equation and see if you can solve for one of the variables and then the other.
We multiplied (x + 2y = 7) by 2 to give us 2x + 4y = 14
Now add the second equation, 2y - 2x = -8
2x + (-2x) + 4y + (2y) = 14 + (-8)
2x - 2x + 4y + 2y = 6
0 + 6y = 6,
Now we can solve for y!
Divide both sides by 6
6y ÷ 6 = 6 ÷ 6
y = 1
Now for the other variable:
Put y = 1 into either of the equations, try x + 2y = 7
x + 2(1) =7
Subtract both sides by 2
x + 2 - 2 = 7 - 2
x = 5
There we have the solution x = 5 and y = 1 (5, 1). Note this is the solution we were given!
This is what is possible when you correctly apply the Properties of Equality to Systems of Equations - we can rearrange and find the solution!
This question will be a walk-through question for solving the system of equations below.
x + 3y = 2
3x - 2y = 6
First step to solving this will be multiplying each equation by a number to make the like terms cancel out. There is a 3y on the top and -2y on the bottom, we can manipulate these along with their equations to cancel them out, to leave us with one variable.
Multiply the top equation by 2 and the bottom by 3.
(x + 3y = 2 )2
(3x - 2y = 6 )3
2x + 6y = 4
9x -6y = 18
Note, 6y and -6y look like they’ll cancel out!
Next, add the two equations together.
+(9x -6y = 18)
2x + 9x + 6y -6y = 18 + 4
11x + 0y = 22
11x = 22
Divide both sides by 11,
11x ÷ 11 = 22 ÷ 11
x = 2
Finally, use the x-value to find the value of y and hence a solution to the system.
Last step to solving this system is to sub in x = 2 into one of our equations (generally, pick the equation that looks like the one you'd be most comfortable solving)
Let’s sub x = 2 into x + 3y = 2
(2) + 3y = 2
2 - 2 + 3y = 2 - 2
3y = 0
Divide both sides by 3
3y ÷ 3 = 0 ÷ 3
y = 0
So, the point (2, 0) is a solution to our system.
2x + y = 7
-3x - 2y = -7
Notice, we have a y in the top equation and a -2y in the bottom, we can multiply the top by 2 to give us -2y and 2y which can cancel out our y terms,
2(2x + y = 7)
2(2x) + 2(y) = 2(7)
4x + 2y = 14
Now add them together
+(-3x - 2y = -7)
4x + (-3x) + 2y + (-2y) = 14 + (-7)
x = 7
Now sub that x value back into one our equations, try 2x + y = 7
2(7) + y = 7
14 + y = 7
Subtract 14 from both sides
14 + y - 14 = 7 - 14
y = -7
So, our solution is, (7, -7) or x = 7 y = -7