Systems of Linear Equations A system of linear equations involves two or more equations. We will be using a system of two linear equations. These systems can be solved graphically, by the substitution method, or by the addition or elimination method.

Systems of Linear Equations A system of linear equations involves two or more equations. We will be using a system of two linear equations. These systems can be solved graphically, by the substitution method, or by the addition or elimination method. Graphically, these systems can have three different outcomes. 1. The two equations will intersect at one point. As you can see, this system of equations intersect at the point( 3 , 2 )

Systems of Linear Equations 2. The two equations are parallel and have no intersection point…

Systems of Linear Equations 3. The two equations are identical and there are infinite solutions. Identical equations occur when one equation can be simplified to equal the other. For example : 𝑥+2𝑦=6 2𝑥+4𝑦=12 ** the second equation is just the first equation multiplied by 2

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point.

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations…

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝒙 𝒚 −1 7 1 −1+𝑦=6 𝑦=7

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝒙 𝒚 −1 7 6 1 −1+𝑦=6 𝑦=7 0+𝑦=6 𝑦=6

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point.Type equation here. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝒙 𝒚 −1 7 6 1 5 −1+𝑦=6 𝑦=7 0+𝑦=6 𝑦=6 1+y=6 𝑦=5

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… Plot the points and graph the line 𝒙 𝒚 −1 7 6 1 5

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝑥+𝑦=6 −3𝑥+𝑦=2 𝒙 𝒚 −1 7 6 1 5 𝒙 −1 1 −3 −1 +𝑦=2 3+𝑦=2 𝑦=−1

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝑥+𝑦=6 −3𝑥+𝑦=2 𝒙 𝒚 −1 7 6 1 5 𝒙 −1 2 1 −3 −1 +𝑦=2 3+𝑦=2 𝑦=−1 −3 0 +𝑦=2 0+𝑦=2 𝑦=2

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝑥+𝑦=6 −3𝑥+𝑦=2 𝒙 𝒚 −1 7 6 1 5 𝒙 −1 2 1 5 −3 1 +𝑦=2 −3+𝑦=2 𝑦=5

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝑥+𝑦=6 −3𝑥+𝑦=2 Plot the points and graph the line 𝒙 𝒚 −1 7 6 1 5 𝒙 −1 2 1 5

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝑥+𝑦=6 −3𝑥+𝑦=2 Plot the points and graph the line 𝒙 𝒚 −1 7 6 1 5 𝒙 −1 2 1 5 Intersection point is ( 1 , 5 )

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝑥+𝑦=6 −3𝑥+𝑦=2 Plot the points and graph the line 𝒙 𝒚 −1 7 6 1 5 𝒙 −1 2 1 5 Intersection point is ( 1 , 5 ) This means that the point ( 1 , 5 ) will satisfy both equations…

Systems of Linear Equations When solving using the graphing method you can either use x / y tables or slope – intercept to graph the linear equations. After they are both graphed, locate the intersection point. EXAMPLE : Find the solution for the linear system by graphing… 𝑥+𝑦=6 −3𝑥+𝑦=2 Create an x / y table for both equations… 𝑥+𝑦=6 −3𝑥+𝑦=2 Plot the points and graph the line 𝒙 𝒚 −1 7 6 𝟏 𝟓 𝒙 −1 2 𝟏 𝟓 Intersection point is ( 1 , 5 ) This means that the point ( 1 , 5 ) will satisfy both equations… Which it does. Luckily it was a point we found for each equation…

Systems of Linear Equations Let’s look at the addition method. Using the addition or elimination method, you want to add or subtract the two equations and eliminate one of the variables making its coefficient equal to zero. This might require you to manipulate one or both of the equations by multiplying the entire equation by some factor to force the elimination to occur. You want to look for or force a plus minus match so the coefficients add up to zero.

Systems of Linear Equations Let’s look at the addition method. Using the addition or elimination method, you want to add or subtract the two equations and eliminate one of the variables making its coefficient equal to zero. This might require you to manipulate one or both of the equations by multiplying the entire equation by some factor to force the elimination to occur. You want to look for or force a plus minus match so the coefficients add up to zero. For example, this system can just be added and the y – variable gets eliminated… 𝑥+2𝑦=10 + 3𝑥−2𝑦=6 4𝑥+0𝑦=16 The y – variable already has a + / - match

Systems of Linear Equations Let’s look at the addition method. Using the addition or elimination method, you want to add or subtract the two equations and eliminate one of the variables making its coefficient equal to zero. This might require you to manipulate one or both of the equations by multiplying the entire equation by some factor to force the elimination to occur. You want to look for or force a plus minus match so the coefficients add up to zero. For example, this system can just be added and the y – variable gets eliminated… 𝑥+2𝑦=10 + 3𝑥−2𝑦=6 4𝑥+0𝑦=16 In this example, we need to multiply the top equation by −3 … 𝑥+2𝑦=10 −3(𝑥+2𝑦=10) −3𝑥−6𝑦=−30 3𝑥−4𝑦=−6 3𝑥−4𝑦=−6 + 3𝑥−4𝑦=−6 0𝑥−10𝑦=−36

Systems of Linear Equations Let’s look at the addition method. Using the addition or elimination method, you want to add or subtract the two equations and eliminate one of the variables making its coefficient equal to zero. This might require you to manipulate one or both of the equations by multiplying the entire equation by some factor to force the elimination to occur. You want to look for or force a plus minus match so the coefficients add up to zero. For example, this system can just be added and the y – variable gets eliminated… 𝑥+2𝑦=10 + 3𝑥−2𝑦=6 4𝑥+0𝑦=16 In this example, we need to multiply the top equation by −3 … 𝑥+2𝑦=10 −3(𝑥+2𝑦=10) −3𝑥−6𝑦=−30 3𝑥−4𝑦=−6 3𝑥−4𝑦=−6 + 3𝑥−4𝑦=−6 0𝑥−10𝑦=−36 ** notice that I picked on the variable that had a coefficient of one. You can also look for multiples…

Systems of Linear Equations In some cases you will need to multiply both equations by a factor to eliminate one of the variables. Look at this example… 2𝑥−4𝑦=13 3𝑥+5𝑦=10

Systems of Linear Equations In some cases you will need to multiply both equations by a factor to eliminate one of the variables. Look at this example… 2𝑥−4𝑦=13 3𝑥+5𝑦=10 I am going to pick on the y – variable because of the different signs.

Systems of Linear Equations In some cases you will need to multiply both equations by a factor to eliminate one of the variables. Look at this example… 2𝑥−4𝑦=13 5(2𝑥−4𝑦=13) 3𝑥+5𝑦=10 4(3𝑥+5𝑦=10) I am going to pick on the y – variable because of the different signs. Multiply each equation by the other equations y – variable coefficient.

Systems of Linear Equations In some cases you will need to multiply both equations by a factor to eliminate one of the variables. Look at this example… 2𝑥−4𝑦=13 5(2𝑥−4𝑦=13) 10𝑥−20𝑦=65 3𝑥+5𝑦=10 4(3𝑥+5𝑦=10) 12𝑥+20𝑦=40 I am going to pick on the y – variable because of the different signs. Multiply each equation by the other equations y – variable coefficient. As you can see, the y – variables coefficient now add up to zero.

Systems of Linear Equations Let’s solve the same system we did with the graphing method using the addition method 𝑥+𝑦=6 −3𝑥+𝑦=2

Systems of Linear Equations Let’s solve the same system we did with the graphing method using the addition method 𝑥+𝑦=6 −3𝑥+𝑦=2 3(𝑥+𝑦=6) Multiplied top equation by 3

Systems of Linear Equations Let’s solve the same system we did with the graphing method using the addition method 𝑥+𝑦=6 −3𝑥+𝑦=2 3(𝑥+𝑦=6) 3𝑥+3𝑦=18 + −3𝑥+𝑦=2 4𝑦=20 Multiplied top equation by 3 Eliminated 𝑥 by addition

Systems of Linear Equations Let’s solve the same system we did with the graphing method using the addition method 𝑥+𝑦=6 −3𝑥+𝑦=2 3(𝑥+𝑦=6) 3𝑥+3𝑦=18 + −3𝑥+𝑦=2 4𝑦=20 𝑦=5 Multiplied top equation by 3 Eliminated 𝑥 by addition Solve for 𝑦

Systems of Linear Equations Let’s solve the same system we did with the graphing method using the addition method 𝑥+𝑦=6 −3𝑥+𝑦=2 3(𝑥+𝑦=6) 3𝑥+3𝑦=18 + −3𝑥+𝑦=2 4𝑦=20 𝑦=5 Now substitute 𝑦=5 into either one of the original equations and solve for 𝑥 Multiplied top equation by 3 Eliminated 𝑥 by addition Solve for 𝑦

Systems of Linear Equations Let’s solve the same system we did with the graphing method using the addition method 𝑥+𝑦=6 −3𝑥+𝑦=2 3(𝑥+𝑦=6) 3𝑥+3𝑦=18 + −3𝑥+𝑦=2 4𝑦=20 𝑦=5 Now substitute 𝑦=5 into either one of the original equations and solve for 𝑥 x+5=6 𝑥=1 SOLUTION is ( 1 , 5 ) Multiplied top equation by 3 Eliminated 𝑥 by addition Solve for 𝑦

Systems of Linear Equations The final method we will look at is called the substitution method. For this method we will solve one of the equations for either x or y getting and then substitute that new algebraic equation into the other equation for that variable.

Systems of Linear Equations The final method we will look at is called the substitution method. For this method we will solve one of the equations for either x or y getting and then substitute that new algebraic equation into the other equation for that variable. Using our original system … 𝑥+𝑦=6 −3𝑥+𝑦=2

Systems of Linear Equations The final method we will look at is called the substitution method. For this method we will solve one of the equations for either x or y getting and then substitute that new algebraic equation into the other equation for that variable. Using our original system … 𝑥+𝑦=6 −3𝑥+𝑦=2 −𝑦=−𝑦 𝑥=6−𝑦 Solved for x in top equation…

Systems of Linear Equations The final method we will look at is called the substitution method. For this method we will solve one of the equations for either x or y getting and then substitute that new algebraic equation into the other equation for that variable. Using our original system … 𝑥+𝑦=6 −3𝑥+𝑦=2 −𝑦=−𝑦 𝑥=6−𝑦 −3 6−𝑦 +𝑦=2 Solved for x in top equation… Substitute that new equation into the other equation for x…

Systems of Linear Equations The final method we will look at is called the substitution method. For this method we will solve one of the equations for either x or y getting and then substitute that new algebraic equation into the other equation for that variable. Using our original system … 𝑥+𝑦=6 −3𝑥+𝑦=2 −𝑦=−𝑦 𝑥=6−𝑦 −3 6−𝑦 +𝑦=2 −18+3𝑦+𝑦=2 −18+4𝑦=2 4𝑦=20 𝑦=5 Solved for x in top equation… Substitute that new equation into the other equation for x… Solved for y …

Systems of Linear Equations The final method we will look at is called the substitution method. For this method we will solve one of the equations for either x or y getting and then substitute that new algebraic equation into the other equation for that variable. Using our original system … 𝑥+𝑦=6 𝑥+𝑦=6 −3𝑥+𝑦=2 𝑥+5=6 𝑥=1 𝑥+𝑦=6 −𝑦=−𝑦 𝑥=6−𝑦 SOLUTION is ( 1 , 5 ) −3 6−𝑦 +𝑦=2 −18+3𝑦+𝑦=2 −18+4𝑦=2 4𝑦=20 𝑦=5 Substitute 𝑦=5 into either equation and solve for x…