Question

Consider the equation 4+2(5x-8)=ax+b for what values of a and b would the equation have infinitely many solutions

Answer 1

To determine the values of a and b for which the equation 4 + 2(5x - 8) = ax + b has infinitely many solutions, we want both sides of the equation to be identical to each other. This way, the equation holds true for all values of x, resulting in the infinite number of solutions.

Let's start by expanding and simplifying the left side of the equation:

4 + 2(5x - 8)

First, distribute the 2 across the terms inside the parentheses:

4 + 2 * 5x - 2 * 8

This simplifies to:

4 + 10x - 16

Now, combine the constant terms:

(4 - 16) + 10x

Which simplifies further to:

-12 + 10x

Now our equation looks like this:

-12 + 10x = ax + b

For the equation to have infinitely many solutions, the expression on the left side must equal the expression on the right side for all values of x. That means the coefficients for x must be the same on both sides, and the constant terms must also be the same.

From the left side of the equation, the coefficient in front of x is 10, and the constant term is -12. So, for the two expressions to be identical:

a must be equal to 10, and b must be equal to -12.

In conclusion, for the equation 4 + 2(5x - 8) = ax + b to have infinitely many solutions, a must be 10 and b must be -12.

Answer 2

Remove the brackets on the left.

4 + 10x - 16 = ax + b

10x - 12 = ax + b

If a = 10 and b = -12 then x can be anything and equality would be true.

So for example let x = 400

10*400 - 12 = 10*400 - 12

4000 - 12 = 4000 - 12

3988 = 3988

You could put in any number you wanted to for x and the results would be equality but meaningless.