This week I learned how to solve word problems using systems, and solving systems by substitution and elimination.

Solving problems using systems:

Step 1: Define the variables (let x=…. and y=…..)

Step 2: Write system of linear equations (translate the word problem, sentence by sentence, into equations)

Step 3: Solve the system (using substitution or elimination)

Step 4: Answer the original question in a full sentence

Example of translating a sentence into equations:

Substitution:

*usually used when 1 variable has a coefficient of 1*

Once you have 2 equations, follow these steps to find the variables:

  • Isolate one of the 4 variables, there are 2 in each equation, it doesn’t matter which one you pick but it’s easier to isolate the one that has a coefficient of 1. This will leave you with one equation that gives you an expression of what the variable is equal to, and one that is still mixed.
  • Plug the expression (of what the variable is equal to) into the other equation in the right spot (if you found what x is equal to then plug it into where x is in the equation)
  • Re-arrange the equation using algebra until you solve for the exact number of what the other variable is equal to. Now you have one of the 2 variables you were looking for.
  • Pick one of the original equations and plug in your first found variable, and solve for your second. You now should have both your variables.
  • Verify: simply plug the numbers you found for the variables into the original equations, if they are true (both sides are equal to each other) then you did everything right, if they don’t match then you should go back and try again as you might have messed up your algebra somewhere

Pro tip: If there ends up being fractions in your equation multiply the entire equation by the denominator (or the greatest common denominator) to get rid of those ugly fractions

Example:

Elimination:

*usually used when all variables have a coefficient that does not equal 1*

Once you have 2 equations, follow these steps to find the variables:

  • Make a zero pair with your 2 equations
  • Add both equations together, leaving you with one variable
  • Isolate the variable, giving you the value of one of the variables
  • Chose one of the original equations and plug in the number into the variable that it corresponds with
  • Isolate the second variable, giving you the value of the last variable you need
  • Verify by plugging in your variable values to see if the equations are true

Pro tip: If there are no zero pairs to start, multiply one of the entire equations by a number that will result in the two equations having a zero pair

Example: