Break even or intersection 6.1 – Systems of equations Types of Solutions: There are 3 types of solutions for linear equations: one solution, no solutions, and infinite solutions One Solution: With equations, one solutions looks like this: 2𝑥+4=𝑥+6 −4 −4 2𝑥 = 𝑥+2 −𝑥 −𝑥 𝑥=2 When you solve for x you get one solution, x = 2 No Solution: With equations, no solutions looks like this: 2𝑥=2𝑥+4 −2𝑥 −2𝑥 0=4 When you solve for x you get no solution, because 0 does not equal 4 Infinite Solutions: With equations, infinite solutions looks like this: 3𝑥+2=3𝑥+2 −2 −2 3𝑥 = 3𝑥 −3𝑥 −3𝑥 0=0 When you solve for x you get a true statement, 0 = 0. This means that x can equal any number so there are infinite solutions Break even or intersection

Using Substitution Method Solving Systems of Equations 𝑦=2𝑥+1 There are 2 variables so we can’t really solve this equation for x or y We need 2 equations to solve Example 1: 𝑦=2𝑥+1 3𝑥−2=𝑦+2 Using Substitution Method 1. Use one of the equations (either one) and solve for y (get y by itself) 𝑦=2𝑥+1 is already solved for y so we will use this one! 2. Substitute the equation solved for y into the other equation Since 𝑦=2𝑥+1, we can substitute 2𝑥+1 for y in the other equation 3𝑥−2= 2𝑥+1 +2 3. Solve for x. 3𝑥−2=2𝑥+3 (add 2) 3𝑥=2𝑥+5 (subtract 2x) 𝑥=5 4. Plug whatever you get in for x into either of the original equations to get y. 𝑦=2 5 +1 so y = 11 Our solution is the point (2 , 11)

Examples Example 2: 10𝑥−𝑦=25 10𝑥−𝑦=40 Example 3: 4𝑥+2𝑦=18 8𝑥+4𝑦=36

Solving Systems of Equations 𝑦=2𝑥+1 There are 2 variables so we can’t really solve this equation for x or y We need 2 equations to solve Example 1: 𝑦=2𝑥+1 3𝑥−2=𝑦+2 Using Graphs Change both equations to slope-intercept form (solve for y, get y by itself) Graph both equations By hand: plot the y-intercept, use the slope to find another point, draw a line Using calculator: hit y=, type one equation into 𝑦 1 and the other into 𝑦 2 3. Find the intersection point Using calculator: hit 2nd , calc, 5, enter, enter, enter Parallel line means no solution If they produce the same line, there are infinite solutions

Examples In your book do pages 380-381 Example 2: 10𝑥−𝑦=25 10𝑥−𝑦=40 4𝑥+2𝑦=18 8𝑥+4𝑦=36 In your book do pages 380-381

6.2 Solving Systems using Linear Combinations AKA Elimination Example 1: 2𝑥+4𝑦=270 3𝑥=435−8𝑦 Using Linear Combinations/ Elimination 1. Get equations in standard form (add and subtract so that x and y are on the same side of the equal sign) and line up the variables 2𝑥+4𝑦=270 is already in standard form 3𝑥=435−8𝑦 we need to add 8y to both sides: 3𝑥+8𝑦=435 2. Look at your coefficients. Find a number that you can multiply one of your equations by so that the coefficients of the same variable are additive inverse (when you add them you get zero) We can multiply the top equation by -2 to eliminate our y variables: −2 2𝑥+4𝑦 =−2(270) −4𝑥−8𝑦=−540 3𝑥+8𝑦=270 3. Add your equations to eliminate one of the variables. Solve for the variable that remains −1𝑥=−105, 𝑥=105 4. Substitute for that variable in one of the original equations to solve for the other variable 2 105 +4𝑦=270 2𝑥+4𝑦=270 3𝑥+8𝑦=435

In your book do pages 388-389, 392-397, 400-402 Sometime you have to multiply both equations by a constant to eliminate one Example 2: 4𝑥+2𝑦=3 5𝑥+3𝑦=4 Example 3: 2.8𝑥=−13.95+5.7𝑦 2𝑦=−9.4−1.4𝑥 In your book do pages 388-389, 392-397, 400-402