Question

PLEASE HELP THIS IS DUE TOMMOROW

Example
Write a constant term or variable term on the line to form an equation that
has no solution, one solution, or infinitely many solutions.
4x + 7 = 4x+
No solution: The x-terms on both sides of the equation are the same.
Write a constant term so the constant terms on each side are different.
4x+7-4x+8
One solution: Write an x-term so the x-terms on each side of the equation
will have different coefficients.
4x+7-4x+14x
Infinitely many solutions: 7 results in identical expressions on both sides
of the equation.
4x+7-4x+7
Look at the Example. Decide whether there is more than one possible answer that
will result in no solution, one solution, or infinitely many solutions. Where possible,
write a different constant term or variable term.
a. No solution: 4x + 7 = 4x+
b. One solution: 4x + 7 = 4x+
c. Infinitely many solutions: 4x + 7 = 4x +

Answer 1

An equation will have no solution if its constant terms are different after the variable terms cancel out, one solution if its variable terms have different coefficients, and infinitely many solutions if both sides are identical after variable terms cancel out.

Step-by-step explanation:

To create an equation that has no solution, one solution, or infinitely many solutions, you should add terms to both sides of the equation so that it either becomes inconsistent, consistent with a unique solution, or consistent with infinitely many solutions.

For no solution, you want the constants on both sides to be different while the variable terms cancel each other out. For example:

4x + 7 ≠ 4x + 5

Here, subtracting 4x from both sides will leave 7 ≠ 5, which is never true; therefore, there is no solution.

For one solution, the variable terms should have different coefficients, thus:

4x + 7 = 4x + 5x - 8

Combining like terms gives us:

4x + 7 = 9x - 8

This equation is solvable for x and will have a unique solution since the coefficients of x are not equal.

For infinitely many solutions, both sides of the equation should be identical after the variable terms are canceled out:

4x + 7 = 4x + 7

Subtracting 4x from both sides leaves 7 = 7, which is always true; hence, the equation has infinitely many solutions because any value of x will satisfy it.