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