Solving Linear Systems by Graphing

A “System” is a set of equations. A Linear System is two or more linear functions (lines) Solving a Linear System Solution= the point of intersection of the lines (where the two lines cross) It is the point at which both functions have the same input with the same output (x,y) If you plug in the x and y values, it should “work” in both equations / make them true

Is (2,-1) a solution to the system? 3x + 2y = 4 Now Check by graphing each equation. Do they cross at (2,-1)? -x + 3y = -5 Plug the (x,y) values in and see if both equations are true. 3(2) + 2(-1) = 4 6 + (-2) = 4 4 = 4 -2 + 3(-1) = -5 -2 + (-3) = -5 -5 = -5 The point works in both equations, so (2,-1) is a solution

Is (2,-1) a solution to the system? y = - 3/2x + 2 3x + 2y = 4 y = 1/3x – 5/3 -x + 3y = -5 Helpful to rewrite the equations in slope-intercept form. Now graph and see where they intersect. The solution is (2,-1)

How Many Solutions? Systems with One Solution Intersecting lines Have equations with different slopes Are called “consistent systems” Systems with Many Solutions / Infinite Soltuions Coincidental Lines =are the same lines / equations ‘in disguise’ Have equations with same slope and same y intercepts Systems with No Solutions Parallel lines Have equations with same slope but different y intercepts Are called “inconsistent systems” y = 3.5x + 65 y = 12x y = -x - 1 y = 2x - 4 y = -2.5x + 7 y = -5/2x + 7 8x - 4y =16 y = 2x - 4 y = 4x + 5 y = 4x - 2 y = 2x + 4 y = 2x - 4

Tip: If in standard form, rewrite in slope-intercept form, then it’s easier to compare the m (slope) and b (y-intercept) values

2x + y = 4 x – y = 2 y = -2x + 4 Slope - 2 y= x - 2 Slope 1 The equations have different slopes Intersecting Lines with One Solution

y = -3x - 1 6x + 2y = -2 Slope -3 y-int. -1 y= -3x - 1 Slope -3 The equations have same slope and same y-intercept Coincidental Lines with Infinitely Many Solutions

12x - 18y = 9 y = 2/3x + 1/2 y = 2/3x - 1/2 Slope 2/3 y-int. - 1/2 The equations have same slope and different y-intercepts Parallel Lines with No Solutions

SOLVE BY GRAPHING -Graph and give solution then check (plug solution into each equation) y = x + 1 y = -x + 5 Solution (2, 3)

2x + y = 4 x – y = 2 y = -2x + 4 y= x - 2 y = x + (-2) y = -2x + 4 Solution: (2,0) y = -2x + 4

Check 2x + y = 4 x – y = 2 2(2) +0 = 4 2 – 0 = 2 4 =4 2 = 2 2(2) +0 = 4 2 – 0 = 2 4 =4 2 = 2 Both equations work with the same solution, so (2,0) is the solution to the system.

Word Problem Example If you invest $9,000 split between two bank accounts, one at 5% and one at 6% interest, and you earn $510 in total interest, how much did you invest in each account? Equation #1 .05x + .06y = 510 Equation #2 x + y = 9,000

(find the x & y-intercepts) Solve by graphing (find the x & y-intercepts) When x = 0 When y = 0 .06y = 510 y = 8,500 y-int (0, 8500) .05x = 510 x = 10,200 x-int (10200 , 0) -------------------------------------------------------------------------------- x + y = 9,000 y = 9,000 y-int (0,9000) x + y = 9,000 x = 9,000 x-int (9000,0)

Investment (3,000, 6,000) Solution 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Thousands at 6% 1 2 3 4 5 6 7 8 9 10 11 Thousands at 5%

CHECK ANSWER TO MAKE SURE!! Graph is upper right quadrant, crossing at (3,000, 6,000) Answer: $3,000 is invested at 5% and $6,000 is invested at 6% CHECK ANSWER TO MAKE SURE!!

Solving Systems by Graphing is often not the easiest or most precise way…