Answer:
(x, y) = (1, 10)
Step-by-step explanation:
The substitution method requires that we write an expression for one of the variables that can be substituted for that variable in the other equation.
Here, the coefficient of y is 1, so it is convenient to write an expression for y.
We choose to use the second equation for that purpose:
2x +y = 12
y = 12 -2x . . . . . subtract 2x from both sides to get an expression for y
Substituting this into the first equation gives ...
5x +(12 -2x) = 15
3x +12 = 15 . . . . . . simplify
x +4 = 5 . . . . . . . divide by 3
x = 1 . . . . . . . . . subtract 4
Using this in our expression for y gives ...
y = 12 -2(1) = 10
The solution is (x, y) = (1, 10).
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Additional comments
In the above solution, we had the 2-step equation 3x+12=15 and we chose to divide by 3. Ordinarily, such a 2-step equation would be solved by subtracting 12 as the first step to give 3x=3. Then the second step would be to divide by 3. We sometimes choose to divide first, "because we can." That is, we recognize that all of the coefficients are multiples of 3, so dividing by 3 leaves us with an equation with smaller coefficients and a multiplier of x that is 1.
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Substitutions can be made for almost anything. Here, there are a couple of interesting opportunities other than the one we used. We can rewrite the first equation a couple of different ways and use the second equation for substitution:
5x +y = 15
3x + (2x +y) = 15
3x +12 = 15 . . . . . . . . substituting for (2x+y)
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5x +y = 15
5x +y = 12 +3
5x +y = (2x +y) +3 . . . . substituting for 12
Note that the effect in these substitutions is to eliminate the y-variable from the equation.
Of course, you can also substitute for x, but that tends to introduce fractions for this problem.
2x +y = 12
x = (12 -y)/2 . . . . . . . . using the second equation to write an expression for x
5((12 -y)/2) +y = 15 . . . . . substituting for x in the first equation
60 -5y +2y = 30 . . . . . . . . . multiply by 2
30 = 3y . . . . . . . add 3y-30 to both sides; simplify
10 = y . . . . . . . divide by 3
