Ch : Solving Systems of Equations Algebraically

What is a System of Linear Equations? If the system of linear equations is going to have a solution, then the solution will be an ordered pair (x (x, y) where x and y make both equations true at the same time. We will only be dealing with systems of two equations using two variables, x and y.y. We will be working with the graphs of linear systems and how to find their solutions graphically. A system of linear equations is simply two or more linear equations using the same variables.

inconsistent consistent dependent independent

Definitions  Consistent: has at least one solution  Dependent: has an infinite amount of solutions

Inconsistent: has no solutions

 Independent: has exactly one solution  Consistent: has at least one solution

x y Consider the following system: x – y = –1 x + 2y = 5 Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they lie on the line. We can also see that any of these points will make the second equation true. However, there is ONE coordinate that makes both true at the same time… (1, 2) The point where they intersect makes both equations true at the same time. How to Use Graphs to Solve Linear Systems

x – y = –1 x + 2y = 5 How to Use Graphs to Solve Linear Systems x y Consider the following system: (1, 2) We must ALWAYS verify that your coordinates actually satisfy both equations. To do this, we substitute the coordinate (1, 2) into both equations. x – y = –1 (1) – (2) = –1  Since (1, 2) makes both equations true, then (1, 2) is the solution to the system of linear equations. x + 2y = 5 (1) + 2(2) = = 5 

Graphing to Solve a Linear System While there are many different ways to graph these equations, we will be using the slope – intercept form. To put the equations in slope intercept form, we must solve both equations for y. Start with 3x + 6y = 15 Subtracting 3x from both sides yields 6y = –3x + 15 Dividing everything by 6 gives us… Similarly, we can add 2x to both sides and then divide everything by 3 in the second equation to get Now, we must graph these two equations Solve the following system by graphing: 3x + 6y = 15 –2x + 3y = –3

Graphing to Solve a Linear System Solve the following system by graphing: 3x + 6y = 15 –2x + 3y = –3 Using the slope intercept forms of these equations, we can graph them carefully on graph paper. x y Start at the y – intercept, then use the slope. Label the solution! (3, 1) Lastly, we need to verify our solution is correct, by substituting (3, 1). Since and, then our solution is correct!

Graphing to Solve a Linear System Let's summarize! There are 4 steps to solving a linear system using a graph. Step 1: Put both equations in slope – intercept form Step 2: Graph both equations on the same coordinate plane Step 3: Estimate where the graphs intersect. Step 4: Check to make sure your solution makes both equations true. Solve both equations for y, so that each equation looks like y = mx + b. Use the slope and y – intercept for each equation in step 1. Be sure to use a ruler and graph paper! This is the solution! LABEL the solution! Substitute the x and y values into both equations to verify the point is a solution to both equations.

Substitution 1. Given two equations. Solve one equation for a variable. 2. Plug this expression in for the variable into the other equation. 3. Solve. 4. Plug this value into the first equation and solve it as well. 5. Write the answer as an ordered pair.

Example: Solve 1. Solve for p 2. Plug into other equation and solve for q. 3. Plug value of q into initial equation. 4. Write the ordered pair (ABC order)

Example: Solve 1. Solve for n 2. Plug into other equation and solve for m. 3. Plug value of m into initial equation. 4. Write the ordered pair (ABC order)

Elimination 1. Given two equations. Make sure variables are lined vertically 2. Choose a variable to eliminate. It must become the opposite value of the same variable in the other equation. 3. Multiply the entire equation to create the needed values. 4. Add the two equations together 5. Solve for variable left. 6. Plug value into initial equation and solve. 7. Write the answer as an ordered pair

Example: Solve 1. Eliminate n since they are opposites. 2. Add the two equations 3. Solve for m 4. Plug m back into the initial equation 5. Solve for n 6. Write the ordered pair (ABC order)

Example: Solve 1. Eliminate h since they are opposites. 2. Multiply the second equation by Add the two equations 4. Solve for g 5. Plug g back into the initial equation 6. Solve for h 7. Write the ordered pair (ABC order)

Ex#1 Solve the system by substitution. Check by graphing – x + 2(-2x + 1) = 2 – x – 4x + 2 = 2 -5x + 2 = x = 0 -5 x = 0 y = -2x + 1 y = -2(0) + 1 y = y = 1 (0, 1) (x, y)

(0, 1)

Ex#2 Solve the system by substitution. 4x – 3(-2x + 13) = 11 4x + 6x – 39 = 11 10x – 39 = x = x = 5 y = -2x + 13 y = -2(5) + 13 y = y = 3 (5, 3) (x, y)

Try These: Solve the system by substitution. 3(2y – 7) + 4y = 9 6y – y = 9 10y – 21 = y = y = 3 x = 2y – 7 x = 2(3) – 7 x = 6 – 7 x = -1 (-1, 3) (x, y)

Ex#2 Solve the system by substitution. 6x – 3(2x – 5) = 15 6x – 6x + 15 = = 15 Infinite solutions

Ex#2 Solve the system by substitution. 6x + 2(-3x + 1) = 5 6x – 6x + 2 = 5 2  5 No Solution

To eliminate a variable line up the variables with x first, then y. Afterward make one of the variables opposite the other. You might have to multiply one or both equations to do this.

Example #3: Find the solution to the system using elimination. 2 6x – 2y = 16 x + 2y = 5 7x + 0 = 21 7x = 21 7 x = y = y = 2 y = 1(x, y) (3, 1) 3x – y = 8 x + 2y = 5

Example #3: Find the solution to the system using elimination x – 3y = –18 4x + 3y = 24 –2x = 6 x = – 3 2(–3) + y = y = 6 y = 12 (x, y) (–3, 12) 2x + y = 6 4x + 3y = 24

3 3x – 4y = 16 5x + 6y = 14 9x – 12y = x = 76 x = 4 (x, y) (4, -1) 5(4) + 6y = y = -6 y = y = 14 10x + 12y = 28 Example #3: Find the solution to the system using elimination.

5 -3x + 2y = -10 5x + 3y = 4 -15x + 10y = y = -38 y = -2 (x, y) (2, -2) 5x + 3(-2) = 4 5x = 10 x = 2 5x – 6 = 4 15x + 9y = 12

Example #3: Find the solution to the system using elimination. 12x – 3y = -9 -4x + y = 3 12x – 3y = = 0 Infinite solutions -12x + 3y = 9

Example #3: Find the solution to the system using elimination. 6x + 15y = x – 5y = 9 6x + 15y = = 15 No solution -6x – 15y = 27