Question

1. francisco made only one mistake on his homework. describe the mistake he made and then solve the equation correctly.
2. quinn and logan solved the equation 8(x - 5)=8x + 40. quinn said the answer was x = 0 and logan said there were no solutions. who is correct? explain your reasoning.
3. nancy wrote two equivalent expressions. if she sets them equal to each other, will the equation have one solution, no solution or infinitely many solutions? use an example to support your answer.

Explanation:

Step1: Identify the mistake

When distributing 3 in $3(x - 7)$, Francisco incorrectly got $3x-7$. It should be $3x-21$ using the distributive property $a(b - c)=ab - ac$ where $a = 3$, $b=x$ and $c = 7$.

Step2: Correctly solve the equation

Start with the correct equation $3(x - 7)=5x - 11$.
Distribute: $3x-21 = 5x - 11$.
Subtract $3x$ from both sides: $3x-3x-21=5x-3x - 11$, which simplifies to $-21 = 2x-11$.
Add 11 to both sides: $-21 + 11=2x-11 + 11$, so $-10 = 2x$.
Divide both sides by 2: $\frac{-10}{2}=\frac{2x}{2}$, and $x=-5$.

Step1: Solve the equation $8(x - 5)=8x + 40$

Distribute on the left - hand side: $8x-40=8x + 40$.
Subtract $8x$ from both sides: $8x-8x-40=8x-8x + 40$.
We get $-40=40$, which is a false statement.

Step1: Consider an example

Let the two equivalent expressions be $2x+3$ and $2x + 3$. If we set them equal to each other: $2x+3=2x + 3$.
Subtract $2x$ from both sides: $2x-2x+3=2x-2x + 3$, we get $3 = 3$. This is a true statement for all values of $x$, so there are infinitely many solutions.

Answer:

Francisco's mistake was incorrect distribution. The correct solution is $x = - 5$.

For question 25: