Rearrange both problems into "y = ax + b" form if possible. For example, "3x -- y = 1" would be rearranged by adding "y" to each side and subtracting one from each side of the equation, providing "y = 3x -- 1."
Solve each equation using the integers from negative five to five as the value of "x." Record the "y" value for each solution.
Plot the "xy" values for both equations on a piece of graph paper. Enter the equations into your graphing calculator if you have one. Draw the lines for both equations.
Find the point on the graph where the two lines intersect. Extend the lines using a straight-edge if they intersect outside of the range you plotted. Record the "x" and "y" values for the intersect point as the solution.
Solve one of the equations for one of the variables. Rearrange the equation into "y = ax + b" using basic algebraic manipulation. For example, "y -- x = 5" could be rearranged to "x = y -- 5" by adding "x" to each side and subtracting five from each side.
Substitute the rearranged equation from above, such as "x = y -- 5," into the other equation. Follow the rules of algebra to solve an equation for one variable. For example, in the system "y -- x = 5 and x + 2y = 8" finding the equation for "x" gives us "x = y -- 5." Replacing "x" in the other equation with "y -- 5" gives us "y -- 5 + 2y = 8" which simplifies to "3y -- 5 = 8." Adding five to both sides of the equation and dividing by three gives "y = 4.33"
Plug the value of the variable you just solved back into the other equation. Solve for the other variable. For example, since "y = 4.33" then "x = y -- 5" equals "x = 4.33 -- 5" or "x = -2/3."
Select a variable to cancel out. Find a variable that has opposing signs in the two equations. For example, if the system is "y --2 x = 5 and 3x + 2y = 8" select the variable "x" to cancel.
Multiply each system by the constants that will give the same coefficient for your chosen variable in either equation. For example, if your system has "2x" in one equation and "3x" in the other, multiply the first by three and the second equation by two so both equations have "6x."
Add the equations together to cancel out the chosen variable. For example, "3y -- 6x = 15 and 6x + 4y = 16" added together equal "7y = 31."
Divide out any coefficient to solve for the single variable. For example "7y = 31" divided by seven equals "y = (31/7)." Plug the value of the known variable back into one of the equations to solve for the other. For example, "(31/7) -- 2x = 5" equals "(33/6) = x" after adding "2x," subtracting 5 and dividing by two.
Create a matrix using the coefficients and constants from the equations. For example, if the system includes "2x -- y = 12" and "x + y = 4," then the matrix would have "2, -1, 12" for the first row and "1, 1, 4" for the second.
Change the upper, left value in the matrix to a one. For example, to change "2" to a "1" simply switch the position of the first and second rows.
Convert the bottom left value in the matrix to a zero. For example, to change the "2" in the matrix with "1, 1, 4" for the first row and "2, -1, 12" for the second row we can add the first row multiplied by negative two to the second row. Multiplying one by negative two gives negative two, which added to two equals zero. Repeating with the middle value gives negative two plus negative one, equaling negative three. Negative two times four equals negative eight, which equals four when added to 12. The new second row equals "0, -3, 4."
Change the bottom middle value in the matrix to a one. For example, divide the bottom row "0, -3, 4" by negative three to get "0, 1, -4/3."
Convert the top middle value to a zero. For example, multiply the second row "0, 1, -4/3" by negative one and add it to the first row "1, 1, 4" to get "1, 0, 16/3" for the top row.
Record the top right value as the "x" value and the bottom right value as the "y" value. For example, with the matrix "1, 0, 16/3" for the first row and "0, 1, -4/3" for the second row, the "x" value is "16/3" and the "y" value is "-4/3."