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.
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