Solving Systems of Equations by Graphing

What is a System of Equations? Solving Linear Systems – The Graphing Method Consistent Systems – one point (x,y) solution Inconsistent Systems – no solution Dependant Systems – infinite solutions Solving Equations Graphically

Concept:A System of Linear Equations Any pair of Linear Equations can be a System

A Solution Point is an ordered pair (x,y) whose values make both equations true

When plotted on the same graph, the solution is the point where the lines cross (intersection)

Some systems do not have a solution

Why Study Systems of Equations?

We will study systems of 2 equations in 2 unknowns (usually x and y)

The algebraic methods we use to solve them will also be useful in higher degree systems that involve quadratic equations or systems of 3 equations in 3 unknowns

A “Break Even Point” ExampleA $50 skateboard costs $12.50 to build, once $15,000 is spent to set up the factory:

Let x = the number of skateboards f(x) = 15000 + 12.5x (total cost equation) g(x) = 50x (total revenue equation)

Using Algebra toCheck a Proposed Solution

Is (3,0) also a solution?

Estimating a Solution usingThe Graphing Method

Graph both equations on the same graph paper If the lines do not intersect, there is no solution If they intersect:

Estimate the coordinates of the intersection point Substitute the x and y values from the (x,y) point

into both original equations to see if they remain true equations

Approximation …Solving Systems Graphically

Practice – Solving by GraphingConsistent: (1,2)

y – x = 1 (0,1) and (-1,0)

y + x = 3 (0,3) and (3,0)

Solution is probably (1,2) …

Check it:

2 – 1 = 1 true

2 + 1 = 3 true

therefore, (1,2) is the solution

(1,2)

Practice – Solving by GraphingInconsistent: no solutions

y = -3x + 5 (0,5) and (3,-4)

y = -3x – 2 (0,-2) and (-2,4)

They look parallel: No solution

Check it:

m1 = m2 = -3

Slopes are equal

therefore it’s an inconsistent system

Practice – Solving by GraphingConsistent: infinite sol’s

3y – 2x = 6 (0,2) and (-3,0)

-12y + 8x = -24 (0,2) and (-3,0)

Looks like a dependant system …

Check it:

divide all terms in the 2nd equation by -4

and it becomes identical to the 1st equation

therefore, consistent, dependant system

(1,2)

Solving Equations by Graphing Equations in one unknown can be split into

two linear equations: 2x + 4 = -2

f(x) = 2x + 4 and g(x) = -2 When the two linear equations are graphed as

a system, the solution is the point (-3,-2) The x-coordinate is the solution to the

original equation in one unknown! The lines above cross at (-3,-2) x = -3

The Downside of Solving by Graphing: It is not Precise

Summary Solve Systems by Graphing Them Together

Graph neatly both lines using x & y intercepts Solution = Point of Intersection (2 Straight Lines) Check by substituting the solution into all equations

Cost and Revenue lines cross at “Break Even Point” A Consistent System has one solution (x,y) An Inconsistent System has no solution

The lines are Parallel (have same slope, different y-intercept)

A Dependent System happens when both equations have the same graph (the lines have same slope and y-intercept)

Graphing can solve equations having one variable