These are three equations and three answers.
Answers:
- A) infinite solutions, identity
- B) one solution, x = 27, neither contradiction nor identiy
- C) x = no solutions, contradiction
Explanation:
Equation A: 20 - 4x = 12 - x + 8 - 3x
The objective is clearing the unknown, which is to isolate it in one side of the equation.
These are the steps and properties that you need to use:
1. Combine like terms on right side:
20 - 4x = 20 - 4x
2. Use addition property of equality (add 4x on each side).
20 = 20 ↔ identity
Conclusion: you have gotten a condition that is always true, no matter the value of x, which means that the equation has infinite solutions and it is an identity.
Equation B: 5x + 4x - 3 = 24 + 8x (note that I placed the equal sign as it was missing)
1. Combine like terms on each side:
9x - 3 = 24 + 8x
2. Subtraction property and addition propertiy of equality (subtract 8x and add + 3 on both sides)
x = 27 ↔ solution (neither contradiction nor identity)
Conclusion: the solution is x = 27.
Equation C: 5x + 6 = 2x + 6 + 3x - 15 (note that I placed the equal signs as it was missing)
1. Comitne like terms on the right side:
5x + 6 = 5x - 9
2. Subtraction property of equality (subtract 5x from each side)
6 = - 9 ↔ contradiction
Conclusion: the equation does not have a solution since the final equality is always false (a condradiction or absurd).
For your information, when the variable (x) dissapears and fhe final equality is always true (for example 0 = 0, or 20 = 20), the equation is an identity.
using back substitution.