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Solving Linear Systems by Graphing

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)

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Solving Linear Systems by Graphing

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  1. Solving Linear Systems by Graphing

  2. 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

  3. Is (2,-1) a solution to the system? Now Check by graphing each equation. Do they cross at (2,-1)? -x + 3y = -5 3x + 2y = 4 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

  4. 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)

  5. How Many Solutions? 8x - 4y =16 y = 2x - 4 y = 3.5x + 65 y = 12x y = 4x + 5 y = 4x - 2 y = 2x + 4 y = 2x - 4 y = -2.5x + 7 y = -5/2x + 7 y = -x - 1 y = 2x - 4 • 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 • Are called “consistent systems” • Systems with No Solutions • Parallel lines • Have equations with same slope but different y intercepts • Are called “inconsistent systems”

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

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

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

  9. y = 2/3x - 1/2 Slope 2/3 y-int. - 1/2 12x - 18y = 9 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

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

  11. y = -2x + 4 2x + y = 4 x – y = 2 y= x - 2 Y y = x + (-2) X 2 -2 Solution: (2,0) y = -2x + 4

  12. Check 2x + y = 4x – y = 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.

  13. 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

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

  15. Investment (3,000, 6,000) Solution 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%

  16. 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!!

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

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