The larger of two positive integers is three more than the smaller. If twice the square of the smaller integer is
increased by three times the larger integer, the result is 74.
Find the integers. [Only an algebraic solution will be
accepted.]

y=x+4

Step-by-step explanation:

So, let's start with the fact that we have two numbers. Let's call one of them "x" and the other one "y." Let's also assume that "y" is the larger number and "x" is the smaller number.

We know that both of them added together (the sum) gives us 74. Mathematically, that means:

x+y=74.

We also know that the larger number ("y") is 4 more than the smaller number ("x"). Another way to say this is, if you have the smaller number and add 4 more to it, you'll end up with the larger number. Mathematically, this means:

y=x+4

Since we know that y=x+4, we can put "x+4" into the first equation for "y", giving us:

x+(x+4)=74

Simplifying gives us:

2x+4=74

   -4   -4

2x=70

---  ---

2    2

x=35

Since we now know that x=35, and we know that y=x+4, we now know that:

y=(35)+4

y=39

Now that we have both of the numbers, we should check our work to ensure it's correct:

Does 39+35=74?      Yes it does.

Is 39 (the larger of the two numbers) four more than the smallest number (35)?    Yes it is.

infinitely many solutions

step-by-step explanation:

we have been given the equation;

2(2x + 3) = -4 + 2(2x + 5)

we need to determine the value of x.

2(2x + 3), we expand this expression to obtain 4x + 6

similarly 2(2x + 5) will become 4x + 10

now we have;

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 10 - 4

4x + 6 = 4x + 6

the expression on the left hand side is identical to the one on the right hand side. therefore, we are looking at just a single straight line whose equation is;

y = 4x + 6

consequently, we shall have infinitely many solutions