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3.03 Equations with variables on both sides

Equations with variables on both sides

When we solve equations with variables on both sides, we will mostly follow all the usual steps that we would when  solving equations  .

The only difference is that we want to rearrange the equation so that all of the variables are on the same side of the equation and all of the constants are on the other side of the equation. This process is similar to grouping like terms.

The side you choose to move variables to does not make a difference to the final solution; however, there is usually a way that will make it easier to solve. For example, if we were solving the equation: 10x - 4 = 6xif we move 6x to the left hand side of the equal sign our x-term is positive. Then we move the -4 to the right hand side and it will also be a positive number.

Examples

Example 1

Solve the following equation: 9x+40=4x

Worked Solution
Create a strategy

Rearrange the equation so that all of the x terms are on the same side.

Apply the idea
\displaystyle 9x+40-4x\displaystyle =\displaystyle 4x-4xSubtract 4x from both sides
\displaystyle 5x+40\displaystyle =\displaystyle 0Simplify
\displaystyle 5x+40-40\displaystyle =\displaystyle 0-40Subtract 40 from both sides
\displaystyle 5x\displaystyle =\displaystyle -40Simplify
\displaystyle x\displaystyle =\displaystyle \dfrac{-40}{5}Divide by 5 on both sides
\displaystyle =\displaystyle -8Simplify the fraction

Example 2

Solve the following equation:

2\left(x-3\right)=x-3

Worked Solution
Create a strategy

Rearrange the equation so that all of the x terms are on the same side.

Apply the idea
\displaystyle 2x-6\displaystyle =\displaystyle x-3Distribute 2 to x-3
\displaystyle 2x-x-6\displaystyle =\displaystyle x-3-xSubtract x from both sides
\displaystyle x-6\displaystyle =\displaystyle -3Simplify
\displaystyle x-6+6\displaystyle =\displaystyle -3+6Add 6 to both sides
\displaystyle x\displaystyle =\displaystyle 3Simplify
Idea summary

To solve an equation with variables on both sides, we can rearrange the equation so that all of the variables are on the same side of the equation and all of the constants are on the other side of the equation.

Number of equation solutions

We are already familiar with equations that have one solution, such as x=4. However, it's also possible for an equation to have either an infinite number of solutions or no solutions.

When equations have exactly one solution, there is only one value that can be substituted for the variable that makes the equation true. The solution of an equation with exactly one solution looks like this:

x=a

When equations have no solutions, there is no value that can be substituted for the variable that makes the equation true. We will get a false statement as a result. The solution of an equation with no solutions looks like this:

a=b(where a and b are different numbers).

When equations have infinitely many solutions, any value can be substituted for the variable and make the equation true. If we try to solve the equation and get a variable or a number equal to itself, then it has infinitely many solutions. For example:

a=a

Examples

Example 3

Consider the equation 18+3x=3x-16.

a

Solve for x

Worked Solution
Create a strategy

Rearrange the equation so that all of the x terms are on the same side and all constant terms on the other side of the equation.

Apply the idea
\displaystyle 18+3x-3x\displaystyle =\displaystyle 3x-16-3xSubtract 3x from both sides
\displaystyle 18\displaystyle =\displaystyle -16Simplify
Reflect and check

Is the statement 18=-16 true?

b

How many solutions does the equation 18+3x=3x-16 have?

Worked Solution
Create a strategy

An equation of the form x=a has one solution.

An equation of the form a=b has no solutions (where a and b are different numbers).

An equation of the form a=a has infinitely many solutions.

Apply the idea

18=-16 is in the form a=b.

The equation 18+3x=3x-16 has no solutions.

Idea summary

An equation of the form x=a has one solution.

An equation of the form a=b has no solutions (where a and b are different numbers).

An equation of the form a=a has infinitely many solutions.

Outcomes

8.EE.C.7

Solve linear equations in one variable.

8.EE.C.7.A

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

8.EE.C.7.B

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

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