How to Solve Systems Using Linear Combinations

Step-by-Step Guide

1. 1

   Step 1: Recognize the standard format.

   In algebra, the “standard format” for an equation is one that is written as Ax+By=C{\displaystyle Ax+By=C}.
   
   When written in this format, the letters A, B and C are commonly chosen to represent numerical values, while x and y are the variables that you need to solve.
   
   You could easily work with different variables, but the structure of the standard format will be the same.
   
   For example, if you are solving a business-related problem about selling hats and scarves to calculate the total number of items sold, you might choose the variable h{\displaystyle h} to represent the number of hats and s{\displaystyle s} to represent the number of scarves.
   
   Your standard format in this case would look like Ah+Bs=T{\displaystyle Ah+Bs=T}.
   
   The steps for solving the problem will still be the same.
2. 2

   Step 2: Rearrange your equations to put them into standard format.

   This may require you to combine similar terms, if each variable appears in the equation more than once, for example.
   
   You will also need to move the terms so they appear in the proper order.
   
   For example, given the equation 2x+y+2y+3x+1=4{\displaystyle 2x+y+2y+3x+1=4}, you need to perform the following steps to get to standard format: 2x+y+2y+3x+1=4{\displaystyle 2x+y+2y+3x+1=4} (given equation) 5x+3y+1=4{\displaystyle 5x+3y+1=4} (combine like terms) 5x+3y=3{\displaystyle 5x+3y=3} (subtract 1 from both sides) You may be familiar with seeing linear equations in the form y=mx+b{\displaystyle y=mx+b}.
   
   This is called the “slope-intercept” form of a line.
   
   It is useful for different purposes.
   
   It could be used to solve the system by linear combinations, but the standard format Ax+By=C is preferred.
   
   If you have your data in the slope-intercept form, you will need to rewrite it algebraically into standard format as follows: y=mx+b{\displaystyle y=mx+b} (given slope-intercept form) y−mx=b{\displaystyle y-mx=b} (subtract mx from both sides)
   -mx+y=b{\displaystyle mx+y=b} (rearrange terms to get x first) A=-m, B=1, C=b (redefine terms for standard format) , It is helpful to write your equations with one directly over the other, so the similar terms line up.
   
   For example, if you have the two equations, in standard format, of 2x−2y=5{\displaystyle 2x-2y=5} and 3x+2y=8{\displaystyle 3x+2y=8}, write them in two rows as: 2x−2y=5{\displaystyle 2x-2y=5} 3x+2y=8{\displaystyle 3x+2y=8}
3. 3

   Step 3: Write your equations so the variables line up.

Detailed Guide

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