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# Solving Equations with Variable on Both Sides

Sep 17, 2022

## 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?

### What have we learned

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

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