System of Equations

2 (or more) equations, each of which has 2 (or more) variables

Our goal is normally to find a the values of the variables that work in both equations.

In most cases you can think of this as the point where two lines cross.

(x,y) is thesolution tothe systemof equations.

There are lots of ways to solve systems of equations.

One way is by graphing. Carefully graph both lines

on the same axes. Find the point where they

cross.

Solve y = 2x + 2y = x – 1

Solve y = 2x + 2y = x – 1

y = 2x + 2 y-intercept = 2 slope = 2/1

y = x – 1 y-intercept = -1 slope = 1/1

The solution is(3, -4)

Solve y = x + 2y = -x + 4

y = x + 2 y-intercept = 2 slope = 1/1

y = -x + 4 y-intercept = 4 slope = -1/1

The solution is(1,3)

Solve x + 2y = 52x + y = 4

Solve x + 2y = 52x + y = 4

Find the intercepts.

Solve x + 2y = 52x + y = 4

Find the intercepts.x + 2y = 5

(0,2.5) and (5,0)2x + y = 4

(0,4) and (2,0)

Thesolutionis(1,2)

Solve 2x + 2y = 64x – 6y = 12

Solve 2x + 2y = 64x – 6y = 12

Intercepts2x + 2y = 6

(0,3) and (3,0)4x – 6y = 12

(0,-2) and (3,0)

We can tell without even finishing the graph that (3,0) is the solution.

Solve 3x + 2y = 123x + 2y = 6

Solve 3x + 2y = 123x + 2y = 6

Intercepts3x + 2y = 12

(0,6) and (4,0)3x + 2y = 6

(0,3) and (2,0)

These areparallel lines.

They both have

a slope or -3/2.

They never intersect, so there is no solution.

It’s also possible to have infinitely many solutions, which would mean you have different expressions for the same line.

For example: 5x + 2y = 10y = -5/2x + 5

There are lots of word problems that involve systems of equations. For instance …

Two tacos and a burrito cost $8. One taco and two burritos cost $10. How much is a taco, and how much is a burrito?

Two tacos and a burrito cost $8. One taco and two burritos cost $10. How much is a taco, and how much is a burrito?

2t + 1b = 8 2x + 1y = 81t + 2b = 10 1x + 2y = 10

2x + 1y = 81x + 2y = 10

The solution is (2,4)

Two tacos and a burrito cost $8. One taco and two burritos cost $10. How much is a taco, and how much is a burrito?

(2,4) means a taco costs $2 and a burrito costs $4.