24. francisco made only one mistake on his homework. describe the mistake he made and then solve the…

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 $x=-5$.

Answer:

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

For question 25:

Explanation:

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.

Answer:

Logan is correct. There are no solutions because the resulting equation $-40 = 40$ is false.

For question 26:

Explanation:

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:

If the two equivalent expressions are truly equivalent, the equation will have infinitely many solutions. For example, if the expressions are $2x+3$ and $2x + 3$, setting them equal gives $3 = 3$, which is true for all $x$.