Applications of Linear Systems Now that you know how to solve a linear system, you can use it to solve real-life problems.

Methods we can use… Graphing- Use this method when after both equations are in slope-intercept form. Graphing- Use this method when after both equations are in slope-intercept form. Substitution-Use this method when one of the variables is isolated. Substitution-Use this method when one of the variables is isolated. Elimination-Use this method when both equations are in Standard Form Elimination-Use this method when both equations are in Standard Form

Selling Shoes A store sold 28 pairs of cross-trainer shoes for a total of $2200. Style A sold for $70 per pair and Style B sold for $90 per pair. How many of each style were sold? A store sold 28 pairs of cross-trainer shoes for a total of $2200. Style A sold for $70 per pair and Style B sold for $90 per pair. How many of each style were sold? Keep in mind that a system has two equations. We need an equation for the quantity of shoes sold and one for the total price. Keep in mind that a system has two equations. We need an equation for the quantity of shoes sold and one for the total price.

Given Information Total number of shoes28 Total number of shoes28 Total receipts2220 Total receipts2220 Price of Style A$70 Price of Style A$70 Price of Style B$90 Price of Style B$90 Assign variables to unknowns Assign variables to unknowns –Number of style Ax –Number of style By

Equation 1: Number of style A + Number of style B = Total number sold Equation 1: Number of style A + Number of style B = Total number sold x + y = 28 x + y = 28 Equation 2: Price A*Quantity A + Price B*Quantity B = Total Price Equation 2: Price A*Quantity A + Price B*Quantity B = Total Price 70x + 90y = x + 90y = 2220

Choose a method & solve I will use substitution… I will use substitution… x + y = 28 x + y = 28 y = 28 – x y = 28 – x Substitute into 2 nd equation Substitute into 2 nd equation 70x + 90*(28 – x) = x + 90*(28 – x) = x x = x x = x = x = -300 x = 15 pairs of Style A x = 15 pairs of Style A

continued y = 28 – x y = 28 – x Substitute x = 15 to find y Substitute x = 15 to find y y = 28 – 15 y = 28 – 15 y = 13 pairs of Style B y = 13 pairs of Style B Solution (15 pairs of Style A, 13 pairs of Style B) Solution (15 pairs of Style A, 13 pairs of Style B)

Mixture Problem Your car’s manual recommends that you use at least 89-octane gasoline. Your car’s 16-gallon gas tank is almost empty. How much regular gasoline (87-octane) do you need to mix with premium gasoline (92-octane) to produce 16 gallons of 89-octane gasoline? Your car’s manual recommends that you use at least 89-octane gasoline. Your car’s 16-gallon gas tank is almost empty. How much regular gasoline (87-octane) do you need to mix with premium gasoline (92-octane) to produce 16 gallons of 89-octane gasoline? You need to know that an octane rating is the percent of isooctane in the gasoline, so 16 gallons of 89-octane gasoline contains 89% of 16, or 14.24, gallons of isooctane. You need to know that an octane rating is the percent of isooctane in the gasoline, so 16 gallons of 89-octane gasoline contains 89% of 16, or 14.24, gallons of isooctane.

Given information Unknowns Unknowns –Volume of regular gasx –Volume of premium gasy Volume of 89-octane16 gallons Volume of 89-octane16 gallons Isooctane in regular.87x Isooctane in regular.87x Isooctane in premium.92y Isooctane in premium.92y Isooctane in 89-octane16*.89 = Isooctane in 89-octane16*.89 = 14.24

Equations Volume of regular + volume of premium = total volume Volume of regular + volume of premium = total volume x + y = 16 x + y = 16 Isooctane in regular + isooctane in premium = Isooctane in 89-octane. Isooctane in regular + isooctane in premium = Isooctane in 89-octane..87x +.92y = x +.92y = 14.24

Solve the system x + y = 16 x + y = 16 y = 16 – x y = 16 – x Substitute into 2 nd equation Substitute into 2 nd equation 0.87x *(16 – x) = x *(16 – x) = x – 0.92x = x – 0.92x = x = x = x = 9.6 gallons of 87 octane x = 9.6 gallons of 87 octane y = 16 – 9.6 = 6.4 gallons of 92 octane y = 16 – 9.6 = 6.4 gallons of 92 octane

Making a decision You are offered two different jobs. Job A offers an annual salary of $30,000 plus a year-end bonus of 1% of your total sales. Job B offers an annual salary of $24,000 plus a year-end bonus of 2% of your total sales. You are offered two different jobs. Job A offers an annual salary of $30,000 plus a year-end bonus of 1% of your total sales. Job B offers an annual salary of $24,000 plus a year-end bonus of 2% of your total sales. How much would you have to sell to earn the same amount in each job? How much would you have to sell to earn the same amount in each job? If you believe you can sell between $500,000 and $800,000 of merchandise per year, which job should you choose? If you believe you can sell between $500,000 and $800,000 of merchandise per year, which job should you choose?

Given information If you pay attention to the wording, the problems gives you an initial amount (b) and percent of sales (m). Both equations can be written in y = mx + b form. If you pay attention to the wording, the problems gives you an initial amount (b) and percent of sales (m). Both equations can be written in y = mx + b form. Job 1: y = 0.01x + 30,000 Job 1: y = 0.01x + 30,000 Job 2: y = 0.02x + 24,000 Job 2: y = 0.02x + 24,000

Solve the system Use your graphing calculator to solve the system since both equations are in slope- intercept form. Use your graphing calculator to solve the system since both equations are in slope- intercept form. Solution (break even point) Solution (break even point) (x = 600,000, y = 36,000) (x = 600,000, y = 36,000) This represents the break even point. This represents the break even point.

continued If you believe you can sell between $500,000 and $800,000 of merchandise per year, which job should you choose? If you believe you can sell between $500,000 and $800,000 of merchandise per year, which job should you choose? If you looked at the graph of the linear system, you can see that if your sales are greater than $600,000, Job B would pay you better than Job A. If you looked at the graph of the linear system, you can see that if your sales are greater than $600,000, Job B would pay you better than Job A.

Assignment Algebra book Algebra book Page 422 Page 422 Problems 31 – 45 odd, Problems 31 – 45 odd,