What are the two numbers?

Question

• seven times the first number plus six times the second number equals 31.

• three times the first number minus ten times the second number is 29.

Answer 1

This question can be solved with two equations, called a system.

Let X be the first number, and Y be the second. Therefore translating this question into math language looks like this:

7X + 6Y = 31 and

3X - 10Y = 29

There are multiple ways to solve this. We can do it via elimination, by substitution, or by graphing both of these lines and finding their intersection point, which would be the answer.

Let's do it by elimination. Our goal is to make the coefficients of one of the variables either equal to each other, or opposites of each other. One way to do this would be to multiply the top equation by 3, and the bottom one by 7.

Doing this looks like:

3(7X + 6Y) = 3(31)

7(3X - 10Y) = 7(29)

After the arithmetic, we get:

21X + 18Y = 93

21X - 70Y = 203

We now subtract the entire second equation from the first, which looks like this:

21X + 18Y = 93

21X -70Y = 203

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_____________

88Y = -110

Y = -5/4

Now that we've found Y, we can substitute it into one of our original equations to find X:

7X + 6(-5/4) = 31

7X + -30/4 = 31

7X = 154/4

X = 5.5

We can verify both of our answers by plugging them both in to the other equation.

3(5.5) - 10(-5/4) = 29

16.5 + 50/4 = 29

16.5 + 12.5 = 29

29 = 29

Since the two sides of the equation are equal to each other, we know our answers are correct. As stated before, another method to check our work would be to graph both equations and find their point of intersection. Sure enough, we see that the two lines intersect at (5.5, -1.25).