Solving Linear Systems by Graphing.

43210 In addition to level 3.0 and above and beyond what was taught in class, the student may: · Make connection with other concepts in math · Make connection with other content areas. The student will write, solve and graph systems of equations and inequalities. - Solve systems of linear equations graphically, with substitution and with elimination method. - Solve systems that have no solutions or many solutions and understand what those solutions mean. - Find where linear and quadratic functions intersect. - Use systems of equations or inequalities to solve real world problems. The student will be able to: - Solve a system graphically. - With help the student will be able to solve a system algebraically. With help from the teacher, the student has partial success with solving a system of linear equations and inequalities. Even with help, the student has no success understanding the concept of systems of equations. Focus 5 Learning Goal – (HS.A-CED.A.3, HS.A-REI.C.5, HS.A-REI.C.6, HS.A- REI.D.11, HS.A-REI.D.12): Students will write, solve and graph linear systems of equations and inequalities.

With an equation, any point on the line (x, y) is called a solution to the equation. With two or more equations, any point that is true for both equations is also a solution to the system.

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

Helpful to rewrite the equations in slope-intercept form. y = - 3 / 2 x +2 y = 1 / 3 x – 5 / 3 Now graph and see where they intersect. Do they cross at (2,-1) ?

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

Solve: If in standard form, rewrite in slope-intercept form, graph the lines, then plug in to check.

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

Check 2x + y = 4 x – y = 2 2(2) +0 = 42 – 0 = 2 4 =4 2 = 2 Both equations work with the same solution, so (2,0) is the solution to the system.

Example 1: If you invest $9,000 at 5% and 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

Solve by graphing (find the x, y- intercepts) When x = 0 When y = 0.06y = 510 y = 8,500 x + y = 9,000 y = 9, x = 510 x = 10,200 x + y = 9,000 x = 9,000

Thousands at 5% Thousands at 6% (3,000, 6,000) Solution Investment

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!!