Question

How do you solve multi step equations with variables on both sides?

Answer 1

Hint: Any equation can be solved by taking all the constants to one side and all the unknowns to the other side of the equation. The constant side must be solved
step-by-step to get through the solution. We need to apply distribution law, addition, subtraction, multiplication and division operations wherever necessary in such a way to simplify the equation.

Complete step by step answer:
There are two methods in solving multi-step equations:

1) Solving equations with one variable on both sides: -

Step-1: We have to apply the distributive property, if necessary. The distributive property is given by \[a(b+c)=ab+ac\]. This rule allows us to cancel out parentheses.
For example: if we have \[2(10-2x)=4(2x+2)\], using distributive law, we get \[\Rightarrow 2(10-2x)=4(2x+2)\to 20-4x=8x+8\]

Step-2: Then, we have to cancel the variable on one side of the equation. To cancel the variate, we have to perform the opposite operation.
For example: If we have \[20-4x=8x+8\], to cancel the \[-4x\] term in the left-hand side of the equation, we have to add \[4x\] on both sides of the equation. That is, we get
\[\Rightarrow 20-4x=8x+8\to 20-4x+4x=8x+8+4x\]

Step-3: Now, we perform addition and subtraction operations wherever necessary.
Example: In the equation \[20-4x+4x=8x+8+4x\], we add and subtract to get
\[\Rightarrow 20-4x+4x=8x+8+4x\to 20=12x+8\]

Step-4: Now, we simplify the equation to get the solution.
Example: In the equation \[20=12x+8\], by subtracting 8 from 20 and then dividing it with 12 we get \[\Rightarrow 20=12x+8\to 12=12x\to x=1\].

Step-5: Verification of solution.
Example: On substituting \[x=1\] back into the equation, we get
\[\Rightarrow 2(10-2x)=4(2x+2)\to 2(10-2(1))=4(2(1)+2)\to 2(10-2)=4(2+2)\to 2(8)=4(4)\to 16=16\]

2) Solving equations with two variables on both sides:

Step-1: we have to isolate one variable in any one of the two equations.
For example, for the equation \[y+1=x-1\], to isolate the y variable, we have to subtract 1 from both the sides. \[\Rightarrow y+1=x-1\to y+1-1=x-1-1\to y=x-2\].

Step-2: We need to substitute this isolated variable into the other equation.
For example, if the first equation is \[2x=20-2y\], then we substitute \[y=x-2\] to make the equation a single variable. \[\Rightarrow 2x=20-2y\to 2x=20-2(x-2)\].

Step-3: Now, we solve for the variable.
For example, \[2x=20-2(x-2)\to 2x=20-2x+4\to 4x=24\to x=6\].

Step-4: Using the value of the variable obtained, we find the other variate.
For example, \[y=x-2\to y=6-2\to y=4\].

Step-5: Verification of the solution.
For example,
\[\begin{align}
  & \Rightarrow 2x=20-2y\to 2(6)=20-2(4)\to 12=20-8\to 12=12 \\
 & \Rightarrow y+1=x-1\to 4+1=6-1\to 5=5 \\
\end{align}\]

These are the different ways in which we solve multi-step equations.

Note: Generally, multi-step equations are more complicated equations. To solve these types of equations, we need to know application of distributive property, addition, subtraction, multiplication and division operations. While simplifying an equation, we generally go wrong when we do calculation mistakes. The mistakes can also occur like writing different values instead of the correct numbers, etc.