9514 1404 393
Answer:
Step-by-step explanation:
Let x and y represent the cost of a pair of shoes and a pair of socks, respectively. Assuming the second purchase was for $128 (not $28), we can describe the purchases by ...
2x +3y = 86
3x +4y = 128
There are many ways to solve a pair of equations like these. Graphing is quick and easy using a graphing calculator. It shows the solution to be ...
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If we want to eliminate the y-variable, we can find the difference of 3 times the second equation and 4 times the first:
3(3x +4y) -4(2x +3y) = 3(128) -4(86)
x = 40 . . . . simplify
Substituting this into the first equation gives ...
2(40) +3y = 86
3y = 6 . . . . . . . subtract 80
y = 2 . . . . . . . . divide by 3
Now, we have the same solution:
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Additional comment
In general, eliminating a variable requires finding the difference of equations where each is multiplied by the coefficient of the variable in the other. Here, the y-coefficients are 3 and 4, so we multiply the first equation by the second coefficient (4) and the second equation by the first coefficient (3). Then we find the difference of those products. Since these are y-coefficients, the difference of y-terms is 0 if done properly, and only the x-variable remains.
Usually, we like to choose multipliers that make this process easier. The numbers can be simpler if the greatest common factor is divided from both multipliers. For example, if the y-coefficients were 3 and 6, then we could divide both multipliers by 3 so they become 1 and 2. Multiplication by 1 is a no-brainer, so already the problem is simpler.
Paying attention to signs and their effect can also simplify the work. If our coefficients were 3 and -4, for example, then our difference could be written as ...
(3)(equation 2) -(-4)(equation 1)
= 3(equation 2) +4(equation 1)
as compared to the equivalent ...
(-4)(equation 1) -(3)(equation 2)
We judge the first version to be simpler to do without making mistakes.
