Several students in Ashton's class were randomly selected and asked how many text messages they sent yesterday. Their answers were 1,0,10,7,13,2,9,15,0,3. How many student were Asked? How do you know?

This is a system of linear equations. We can solve by elimination. First, we write out the two equations:

3m - 4n = 1

4m - 6n = 5

We want to make the coefficients of one of the variables the same so that we can eliminate it. Let's make the coefficient of m 12 for both equations. To do that for the first, we have to multiply both sides by 4:

12m - 16n = 4

To do that for the second, we have to multiply both sides by 3:

12m - 18n = 15

Put the two equations together:

12m - 16n = 4

12m - 18n = 15

We can subtract the bottom equation from the top and eliminate the 12m to get:

-16n - (-18n) = 4 - 15

2n = -11

n = -5.5

Now that we know n, we can solve for m. We can use one of the original equations and put in the value -5.5 for n and solve for m: 3m - 4 * (-5.5) = 1

3m + 22 = 1

m = -7

and

n = -5.5

3m - 4n = 1​ Equation 1

4m - 6n = 5​ Equation 2

We clear the m variables in the two equations

m = (1+4n)/3 Equation 1

m = (5 +6n)/4 Equation 2

Now we match them

(1+4n)/3 = (5+6n)/4

4(1+4n) = 3(5+6n)

4+16n = 15 +18n

16n - 18n = 15 -4

-2n = 11

n = -11/2

Now we substitute this value in one of the equations

4m -6n = 5 Equation 2

4m -6(-11/2) = 5

4m +66/2 = 5

4m + 33 = 5

m = (5 -33)/4

m = -28/4 = -7

Answer m = -7 and n = -11/2

checking

3*(-7) - 4(-11/2) = 1

-21 +44/2 = 1

-21 +22 = 1 Correct !

4(-7) -6(-11/2) = 5

-28 +66/2 = 5

-28 +33 = 5 Correct!

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