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It doesn't matter which equation you decide to work with or even which variable you choose to solve for, as you should find the same solution no matter what. However, you do want to make the process as simple as possible. You should choose the equation that you think will be easiest to work with. For instance, if there is an equation where one of the coefficients is 1, such as x - 3y = 7, you would choose that as it will be easy to solve for 'x'. For example, let's say our equations are: x - 2y = 10 (equation A) and -3x -4y = 10 (equation B). You would choose to work with x - 2y = 10 because the coefficient of x in this equation is 1. Solving for x in equation A would meaning adding 2y to both sides. Therefore, x = 10 + 2y. For this step, you will need to insert (or substitute) your solution for 'x' into the other solution that you did not work with. This will allow you to find the other variable, in this case 'y'. Let's try it: Insert the 'x' of equation B into equation A: -3(10 + 2y) -4y = 10. You can see that we have taken 'x' out of the equation and inserted what 'x' equals. Now that you have removed one of the variables from the equation, you can solve for the other variable. This is is simply solving a regular one-variable linear equation. Let's solve ours: -3(10 + 2y) -4y = 10 so -30 -6y -4y = 10. Combine the y's: -30 - 10y = 10. Move the -30 over to the other side: -10y = 40. Solve for y: y = -4. To do this, plug your findings for 'y', or the first variable, into one of the equations. Then solve for the other variable, in this case 'x'. Let's try it: Solve for 'x' in equation A by plugging in y = -4: x - 2(-4) = 10. Simply the equation: x + 8 = 10. Solve for x: x = 2. Plug both of the variables into each equation to make sure that they create true equations. Let's see if ours work: Equation A: 2 - 2(-4) = 10 is TRUE. Equation B: -3(2) -4(-4) = 10 is TRUE.
Begin by solving one equation for either variable. Substitute your findings in Step 1 into the other equation. Solve for the other variable. Solve the second variable. Double check that the variables you have found work for both equations.