- 1). Rearrange the terms in both equations, if necessary, so that the "x" and "y" terms appear on the left-hand side of the equals sign and the constant term appears on the right-hand side of the equals sign, with "x" terms appearing before "y" terms. For instance, say that the two equations you are given are 3y = -2x + 17 and 5y + 3x = 27. In the first equation, add 2x to both sides, and in the second equation, switch the order of the 5y and the 3x. Hence, the system of equations becomes 2x + 3y = 17 and 3x + 5y = 27.
- 2). Write the reordered equations vertically, with one equation directly underneath the other, so that their "x" terms, "y" terms, equals signs and constant terms appear directly above and below one another. It doesn't matter which equation is written on top and which is written on the bottom. Draw a set of parentheses around both equations.
- 3). Decide whether you intend to eliminate the "x" terms or the "y" terms. This is a matter of both personal preference and logic. You can choose to eliminate either set of terms and still arrive at the correct answer; however, it's simpler to eliminate the terms that possess a lower-least common multiple, or LCM, because then you can perform calculations on smaller numbers. In 2x + 3y = 17 and 3x + 5y = 27, eliminating the "x" terms will be slightly simpler because their LCM is only 6, whereas the LCM of the "y" terms is 15.
- 4). Determine what number you need to multiply the first, or top, equation by to produce the LCM of the selected variable's coefficient. In 2x + 3y = 17, you need to figure out which number to multiply the "2" by to produce the LCM of the "x" terms, which is 6. In this equation, you'll need to multiply by 3 because 2*3 = 6. Write this number to the left of the parentheses: 3 (2x + 3y = 17).
- 5). Determine what number you need to multiply the second, or bottom, equation by to obtain the LCM of the coefficient of the selected variable. In 3x + 5y = 27, you want to find which number to multiply the "3" by to obtain the LCM, 6. In this equation, you'll need to multiply by 2, because 3*2 = 6. Write this number to the left of the parentheses: 2 (3x + 5y = 27).
- 6). Perform multiplication on both equations by multiplying the constant outside each set of parentheses by every single term inside the parentheses. In 3 (2x + 3y = 17), multiply 3*2x to get 6x, multiply 3*3y to get 9y and multiply 3*17 to get 51, resulting in a new equation: 6x + 9y = 51. In 2 (3x + 5y = 27), multiply 2*3x to get 6x, multiply 2*5y to get 10y and multiply 2*27 to get 54, producing a new equation: 6x + 10y = 54. In these two new equations, 6x + 9y = 51 and 6x + 10y = 54, the coefficients of the "x" terms now match, and you can solve the system via subtraction. If you had chosen to eliminate the "y" terms, their coefficients would now match, and you would still proceed to solve through subtraction.
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