5.04 Variables on both sides
Introduction
When solving equations we often aim to isolate the variable or pronumeral we are solving for. We also know that if we have multiple like terms on one side of the equation we can combine them into a single term.
So what happens if we have variables on both sides of the equation?
Move the variables
In the same way that we apply reverse operations to isolate our variables, we can also use them to gather variable terms together.
Whenever we want to solve equations that have variables on both sides, our main goal is to move all the variable terms to one side so that we can combine them, then use reverse operations to finish solving the equation.
While the side we move the variable terms to doesn't affect the final answer, it can introduce a lot of negative signs that will need to be cancelled out later. For this reason, it is best to move the smaller variable term over as to avoid negative coefficients.
Examples
Example 1
Solve the equation: 4x+24=x-9
Whenever we want to solve equations that have variables on both sides, our main goal is to move all the variable terms to one side so that we can combine them, then use reverse operations to finish solving the equation.
Equations with brackets and variables on both sides
Suppose we want to solve the equation:8(6x-4)=7(-8x+6)+134
We have a couple of choices for our first step; we can either apply the distributive law to expand all the brackets in the equation, or we can move the whole expression 7(-8x+6) to the left hand side.
If we choose to apply the distributive law first, the working out to solve the equation will look like this:
| \displaystyle 8(6x-4) | \displaystyle = | \displaystyle 7(-8x+6)+134 | |
| \displaystyle 48x-32 | \displaystyle = | \displaystyle -56x + 42 +134 | Use the distributive law to expand the brackets |
| \displaystyle 48x+56x-32 | \displaystyle = | \displaystyle 42+134 | Reverse the subtraction of 56x |
| \displaystyle 104x-32 | \displaystyle = | \displaystyle 42+134 | Combine the x-terms together |
| \displaystyle 104x | \displaystyle = | \displaystyle 42+134+32 | Reverse the subtraction |
| \displaystyle 104x | \displaystyle = | \displaystyle 208 | Perform the addition |
| \displaystyle x | \displaystyle = | \displaystyle 2 | Reverse the multiplication |
If we choose to move all the variables to the left hand side first, the working out to solve the equation will look like this:
| \displaystyle 8(6x-4) | \displaystyle = | \displaystyle 7(-8x+6)+134 | |
| \displaystyle 8(6x-4) - 7(-8x+6) | \displaystyle = | \displaystyle 7(-8x+6)+134 - 7(-8x+6) | Subtract 7(-8x+6) from both sides |
| \displaystyle 48x-32+56x-42 | \displaystyle = | \displaystyle 134 | Use the distributive law |
| \displaystyle 104x-32-42 | \displaystyle = | \displaystyle 134 | Combine the x-terms |
| \displaystyle 104x-74 | \displaystyle = | \displaystyle 134 | Combine the constant terms |
| \displaystyle 104x | \displaystyle = | \displaystyle 208 | Reverse the subtraction |
| \displaystyle x | \displaystyle = | \displaystyle 2 | Reverse the multiplication |
As we can see, both methods take the same number of steps and are both about as difficult as each other. We could also choose to move the expression 8(6x-4) to the right hand side as our first step which would involve almost identical steps to the working shown directly above.
Examples
Example 2
Solve this equation for x by first expanding the brackets:8(8x+5)=3(6x+8)+108
To solve equations with brackets and variables on both sides we can either apply the distributive law to expand all the brackets in the equation first, or we can move one set of brackets to the other side first so that the variables are on the same side.
Cross-multiply to solve
It should also be noted that for equations like:\dfrac{6x+4}{8 }=\dfrac{-3x+12}{3}
The division of the numerators by the denominators will need to be reversed first. The result of this is that each numerator gets multiplied by the denominator from the other side of the equation, like so:3(6x+4)=8(-3x=12)
This is called cross-multiplication.
Examples
Example 3
Solve this equation for x by cross-multiplying and then expanding the brackets:\dfrac{5x-4}{3 }=\dfrac{3x+4}{5}
When cross multiplying fractions, we remove the denominators by multiplying them to the opposite numerators, as shown below:
Which gives us:
After performing this step, we can then distribute or move the variables to one side and solve the equation.