Solving Equations with Variable on Both Sides

Key Concepts

• Solving equations with variable on both sides
• Solving equation with infinitely many or no solutions
• Use equations to solve problems

Solving algebraic equations 

If an equation has pronumerals on both sides, collect the like terms to one side by adding or subtracting terms. 

Example:

2x+1 = 5x−4

1 = 3x−4 [Subtracting 2x from both sides] 

3x = 5 [Adding 4 to both sides] 

x = 5/3 [Dividing by 3] 

Solve equations using algebra tiles 

Example: Solve for 4x + 7 = 2(2x + 1) + 5 

Solution: Let us represent the equation using algebra tiles 

Since 7=7 is a true statement, the equation is true for all values of x. 

So, x can have infinitely many solutions. 

Identity 

An equation that is true for all values of the variable is an identity. 

Q: Solve 4x – 3 = 2(2x – 3) + 3 

Sol: 4x – 3 = 4x – 6 + 3 

        4x – 3 = 4x – 3 

              – 3 = – 3 

Equation that has no solution 

Q: Find the value of x if 3(x + 2) – 7 + 2x = 5x + 4. 

Sol: 3(x + 2) – 7 + 2x = 5x + 4 

    3x + 6 – 7 + 2x = 5x + 4      [Distributive property] 

   (3x + 2x) + (6 – 7) = 5x + 4       [Add like terms] 

                  5x – 1 = 5x + 4                 [Maintain the equality by subtracting 5x from both sides] 

                        – 1 = 4 

There is no value of x that makes the equation true. 

Therefore, the equation has no solution. 

Exercise

• Solve 5x-4 = 4x
• The triangle shown is isosceles. Find the length of the third side of the triangle.

• Solve the equation – 3(8+3h) = 5h+4
• Find the missing value in – 2(2x- ?) + 1 = 17-4x
• Is the equation – 4(3-2x) = -12-8x an identity?

Concept Map 

What have we learned

• If an equation has pronumerals on both sides, collect the like terms to one side by adding or subtracting terms.