Solving Equations With Variables On Both Sides [pdf] is a fundamental skill for students. We have seen how to solve equations when variable is on one side. Now we will see how to solve equations when variables are on both sides.
We have algebraic equations that have been carefully generated with the unknown or variable present either on the left or right side of the equation.
But, what if an algebraic equation has variable terms present on both sides of the equal sign? For example, consider this equation:
Students are often faced with a problem when they need to solve an equation with variables on both sides. Variables are symbols that stand for unknown quantities. Variables can be represented by letters, such as x, y, and z. This can be solved by using the following steps:
Steps for Solving Equations With Variables On Both Sides
1) Solve for the variable on one side of the equation first.
2) Substitute this value into the other side of the equation and solve.
Here, the variable term, 50x, appears on the left side of the equation. and another variable term, 60x, appears on the right side of the equation. Before isolating the variable, you must first get the variable terms on one side of the equation and the constant terms on the other side.
You have two approaches: you can either
1) add 50x to both sides of the equation to eliminate the – 50x on the left side of the equation and obtain
50x + 220 – 50x = 60x + 50x
which reduces to
220 = 110x
2) subtract 60x from both sides of the equation to eliminate the 60x on the right side of the equation to obtain
-60x + 220 – 50x = 60x – 60x
which reduces to
-110x + 220 = 0
Comparing Both Approaches
Of the above approaches approach (1) facilitates a generates faster solution since adding 50x to both sides of the equation accomplishes having only the constant 220 on the left side of the equation and the variable term on the right side of the equation: 220 = 110x.
To solve for x here we did inverse operation and divide both sides of the equation by 110 to obtain x = 2.
Using approach (2), to solve – 110x + 220 = 0, we isolate the variable by doing the inverse operation and first subtracting 220 from both sides of the equation, yielding
-110x = –220.
Next, since x is multiplied by – 110, you do the inverse and divide both sides of the equation by – 110 to obtain the solution, x = 2.
Notice that regardless of the approach you choose—removing the variable term from the left side of the equation or removing the variable term from the right
side of the equation—you derive the same solution;
However, the route to the solution is faster by taking the approach that leaves the constant on one side and the variable on the other side (as was the case using approach 1 above).
Consider another example of an equation with the variable on both sides,
3(x – 5) + 3 = 4(-2x + 3) + 7x
You first use the distributive property to obtain
3x – 15 + 3 = – 8x + 12 + 7x
and then combine like terms, yielding the equivalent equation,
3x – 12 = -x + 12
At this point, you have the option of either eliminating (1) the variable term –x on the right side of the equation by adding x to each side of the equation or (2) the variable term 3x on the left side of the equation by subtracting 3x from each side of the equation.
Should you choose approach 1or approach 2?
It does not matter, but simply due to a preference to work with positive variable terms, you will add x to each side of the equation, thus you now have a typical two-step equation,
4x – 12 = 12
Since the left side of the equation contains the variable term and 12 is subtracted from this term, you do the inverse and add 12 to each side of the equation,
4x = 24
Finally, since x is multiplied by 4, you do the inverse and divide each side of the equation by 4, yielding the solution,
x = 6
Had you taken the alternate approach and subtracted 3x from the equation 3x – 12 = -x + 12, you would have obtained the two-step equation,
-12 = -4x + 12.
Since the right side of the equation contains the variable term, we must first subtract 12 from each side of the equation to obtain -24 = -4x, then divide each side of the equation by -4.