​What is a linear system?

​A linear system is a single Cartesian plane with multiple lines graphed on it. These lines sometimes intersect and their intersection is called the solution of the linear system. If the lines are parallel, then they do not intersect and there is no solution. Sometimes, the same line is given twice, and there are infinitely many solutions.

​Method 1: Graphing

​We can solve linear systems by graphing them and finding the point at which they intersect. In order to graph linear equations in standard form, we need to convert them to slope-intercept form (to ensure the lines aren't parallel or the same line). If the slopes are the same, the lines are parallel and there is no solution. If the slopes and y-intercepts are the same, then they are the same line and there are infinitely many solutions.

​Other Practice Problems

Find the solution for the following linear systems:​

7x−8y=−12

−4x+2y=3

​

​3x+9y=−6

−4x−12y=8

​

​x−7y=−1

15x+2y=−18

​Other Practice Problems

Find the solution for the following linear systems:​

7x−8y=−12

−4x+2y=3 (0, 3/2)

​

​3x+9y=−6

−4x−12y=8 Infinitely Many Solutions

​

​x−7y=−1

15x+2y=−18 (-4, 1)

​Method 2: Substitution

​We can also solve linear systems using algebra. The first algebraic method for solving linear systems is substitution. In this method, we solve one of the equations given for either x or y. Once we do, we can plug in that equation to the value of the solved variable. Once we solve for the other variable that's left, we can use that value to solve for the other value.

​Steps for Substitution

Solve the following system:

​6x−5y=8

−12x+2y=0

(Step 1) Solve for y in the second equation.

-12x + 2y = 0

+12x +12x

2y = 12x

/2 /2

y = 6x

​Steps for Substitution

Solve the following system:

​6x−5y=8

−12x+2y=0

​(Step 2) Plug the value of y in to the other equation and solve.

6x - 5(6x) = 8

​6x - 30x = 8

​-24x = 8

​/-24 /-24

​x = -1/3

​Steps for Substitution

Solve the following system:

​6x−5y=8

−12x+2y=0

​(Step 3) Plug in the found value of x to either equation and solve for y.

​6(-1/3) - 5y = 8

​-2 - 5y = 8

​+2 +2

​ -5y = 10

​ /-5 /-5

​ y = -2

​Steps for Substitution

Solve the following system:

​6x−5y=8

−12x+2y=0

​(Step 4) Write the answer as a coordinate pair (x, y).

​(-1/3, -2)

​Method 3: Elimination

​For elimination, we match up the coefficient of the same variable in each equation usually by means of multiplication. Once they are matched up, we either add or subtract in order to cancel that variable out, allowing us to solve for the variable that is left.

​Steps for Elimination

Solve the following linear system:

​−2x+10y=25

5x−25y=3

​(Step 1) Multiply each equation to match up a variable's coefficient. Let's match the x coefficients.

​(-2x + 10y = 25) x 5 -> -10x + 50y = 150

​(5x - 25y = 3) x 2 -> 10x - 50y = 6

​Steps for Elimination

Solve the following linear system:

​−2x+10y=25

5x−25y=3

​(Step 2) Since the coefficients of x have opposite signs, we can add the two equations to each other in order to cancel the x variable.

​-10x + 50y = 150

​+(10x - 50y = 6)

​0x + 0y = 156

​0 = 156 <- Since this statement is false, there is no solution.

​Practice!

Solve the following linear systems:​

2x+3y=20

7x+2y=53

​

​

​3x - y = 7

​2x + y = 8