You can put this solution on YOUR website! 2x+4y=40
7x-3y=4
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I am going to call 2x and 4y the variable terms in the top equation and 7x and 3y the variable
terms in the bottom equation.
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We can't make the elimination method work unless we get one of the variable terms in the
top equation to equal its counterpart in the bottom equation. We make that happen by multiplying
both sides of one of the equations by one number and then multiplying both sides of the
other by a different number. Then we can add or subtract the two equations to make the common
term disappear.
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Before we do anything else, let's do that much on your problem so you can see what that means.
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You were given the two equations:
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2x+4y=40
7x-3y=4
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We could eliminate the y terms if we wanted to, but in this case we are going to eliminate
the x terms in the two equations. Let's begin by multiplying all the terms in the top equation
by 7. If we do that the top equation becomes:
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14x + 28y = 280 and the bottom equation stays the same
7x - 3y = 4
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Now let's multiply the all the terms in the bottom equation by 2. When we do that the
pair of equations becomes:
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14x + 28y = 280 and
14x - 6y = 8
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Can you now see why we chose to multiply the top equation by 7 and the bottom equation
by 2? We did so because those numbers made the x term in each equation the same value.
Now we can subtract (in columns) the two equations.
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When we subtract the 14x in the bottom equation from the 14x in the top equation, the
result is 0 so the x term has been eliminated. Next we subtract -6y in the bottom
equation from +28y in the top equation. This subtraction is [28y - (-6y)] which simplifies
to [28y + 6y] and the answer is 34y. Finally, on the right side we subtract 8 from 280 and
the answer to that is 272. The result of these subtractions is shown below:
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14x + 28y = 280
14x - 6y = 8
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0 + 34y = 272 <---- this is the resulting new equation
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Now we can solve for y by dividing both sides of the resulting equation by 34. When we do
that division we get:
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After completing this division, the left side becomes just y and the right side is 8.
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So we now know that y = 8. We can then return to either of the original equations,
substitute 8 for y and solve the equation to get the value of x.
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Let's return to the top equation:
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2x + 4y = 40
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Substituting 8 for y makes this equation:
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2x + (4*8) = 40
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Multiply the numbers in parentheses on the left side and you get:
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2x + 32 = 40
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Get rid of the 32 on the left side by subtracting 32 from both sides to get:
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2x = 8
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And divide both sides of this equation by 2 to find that:
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x = 8/2 = 4
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So the answers to your problem are x = 4 and y = 8
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And you can further check this by substituting these two numbers into the original
bottom equation to see if both sides of that equation remain equal.
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Hope this helps you to understand the process of elimination when applied to two linear
equations. The whole trick to this process is choosing multipliers for the two equations
that will make one variable term in the top equation equal it corresponding variable
term in the bottom equation.
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