Systems of Equations & Inequalities Mrs. Daniel Algebra 1

Table of Contents 1.What is a System? 2.Solving Systems Using Substitution 3.Solving Systems Using Elimination 4.Solving Systems Using Graphing 5.Systems & Word Problems 6.Solving Systems of Inequalities by Graphing

What is a System?! System of Linear Equations - a set of two or more linear equations containing two or more variables. – Example: Solution of a System of Linear Equations – is an ordered pair that satisfies each equation in the system.

Three Methods for Solving Systems Graphing Substitution Elimination Method selected depends on information provided and personal preference.

Possible Types of Solutions One point of intersection, so one solution. Parallel lines, so no solution. Same lines, so infinite # of solutions.

If an ordered pair does not satisfy the first equation in the system, there is no reason to check the other equations. Helpful Hint Solutions We check to determine if a solution is valid by plugging in x or y to EACH equation. Must satisfy both equations to be a valid solution.

Tell whether the ordered pair is a solution of the given system. (–2, 2); –x + y = 2 x + 3y = 4 Let’s Practice….

Tell whether the ordered pair is a solution of the given system. (1, 3); 2x + y = 5 –2x + y = 1 Let’s Practice….

Tell whether the ordered pair is a solution of the given system. (2, –1); x – 2y = 4 3x + y = 6 Let’s Practice….

Wren and Jenni are reading the same book. Wren is on page 14 and reads 2 pages every night. Jenni is on page 6 and reads 3 pages every night. After how many nights will they have read the same number of pages? How many pages will that be? Writing a System of Equations

Video club A charges $10 for membership and $3 per movie rental. Video club B charges $15 for membership and $2 per movie rental. For how many movie rentals will the cost be the same at both video clubs? What is that cost? Let’s Practice…

Solving Systems Using Substitution

Theory of Substitution Method Substitution is used to reduce the system to one equation that has only one variable. Then you can solve this equation for the one variable and substitute again to find the other variable.

Solving Systems of Equations by Substitution Step 2 Step 3 Step 4 Step 5 Step 1 Solve for one variable in at least one equation, if necessary. Substitute the resulting expression into the other equation. Solve that equation to get the value of the first variable. Substitute that value into one of the original equations and solve for the other variable. Write the values from steps 3 and 4 as an ordered pair, (x, y), and check. How to: Substitution Method

Solve the system by substitution. y = 3x y = x – 2 Example: Substitution

Solve the system by substitution. y = x + 1 4x + y = 6 Let’s Practice…

Solve the system by substitution. x + 2y = –1 x – y = 5 Let’s Practice…

Solving Systems by Elimination

Since –2y and 2y have opposite coefficients, the y-term is eliminated. The result is one equation that has only one variable: 6x = –18. Theory of Elimination Method

How to: Elimination Method Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Step 3: Add or subtract the equations. Step 4: Plug back in to find the other variable. Step 5: Check your solution. Standard Form: Ax + By = C Look for variables that have the same coefficient. Solve for the variable. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations.

Let’s Practice… x + y = 5 3x – y = 7 Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. They already are! The y’s have the same coefficient. Step 3: Add or subtract the equations. Add to eliminate y. x + y = 5 (+) 3x – y = 7

Step 4: Plug back in to find the other variable. Step 5: Check your solution. x + y = 5 3x – y = 7 Let’s Practice…

y + 3x = –2 2y – 3x = 14 Solve by elimination. Let’s Practice…

3x – 4y = 10 x + 4y = –2 Solve by elimination. Let’s Practice…

Example: Elimination by Subtracting 4x + y = 7 4x – 2y = -2 Step 1: Put the equations in Standard Form. They already are! Step 2: Determine which variable to eliminate. The x’s have the same coefficient. Step 3: Add or subtract the equations. Subtract to eliminate x. 4x + y = 7 (-) 4x – 2y = -2

Step 4: Plug back in to find the other variable. Step 5: Check your solution. 4x + y = 7 4x – 2y = -2 Example: Elimination by Subtracting

3x + 3y = 15 –2x + 3y = –5 Solve by elimination. Let’s Practice…

2x + y = –5 2x – 5y = 13 Solve by elimination. Let’s Practice…

When Elimination Is Not Obvious You can use the elimination method, even if the coefficients are not the same. Just multiply all terms in one equation by the LCM to make the same.

2x + 2y = 6 3x – y = 5 Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. They already are! None of the coefficients are the same! Find the least common multiple of each variable. Which Variable is Easier?? _________________________ More Elimination

Step 4: Plug back in to find the other variable. 2x + 2y = 6 3x – y = 5 Step 3: Multiply the equations and solve. Multiply the bottom equation by 2 More Elimination Step 5: Check your solution.

x + 2y = 11 –3x + y = –5 Solve the system by elimination. More Elimination

x + 2y = 11 –3x + y = –5 Solve the system using the method of your choice. Let’s Practice…

Solve the system using the method of your choice. 3x + 2y = 6 –x + y = –2 Let’s Practice…

Solving Systems by Graphing

Steps for Obtaining the Solution of a System of Linear Equations by Graphing Step 1: Graph the first equation in the system. Step 2:Graph the second equation in the system. Step 3:Determine the point of intersection, if any. Step 4: Verify that the point of intersection determined in Step 3 is a solution of the system. Remember to check the point in both equations. Solving a System by Graphing

Solve the system by graphing. Check your answer. y = x y = –2x – 3 Let’s Practice…

Solve the system by graphing. Check your answer. y = –2x – 1 y = x + 5 Let’s Practice…

Solve the system by graphing. Check your answer. Let’s Practice… 2x + y = 4

Systems & Word Problems

Unpacking Word Problems 1. Identify each variable. – Think: “What does each variable represent?” 2. Set up two equations. 3. Determine your method of solution. – Think: “How can I solve the system in the easiest way possible?” 4. Solve the system. 5. Check your answer!

Hints!!!!

Jenna is deciding between two cell-phone plans. The first plan has a $50 sign-up fee and costs $20 per month. The second plan has a $30 sign-up fee and costs $25 per month. After how many months will the total costs be the same? What will the costs be? If Jenna has to sign a one-year contract, which plan will be cheaper? Explain. Let’s Practice…

One cable television provider has a $60 setup fee and $80 per month, and the second has a $160 equipment fee and $70 per month. In how many months will the cost be the same? What will that cost be? Let’s Practice…

The population of Sunny Isles is 50,000 but is growing at 2500 people per year. North Miami has a population of 26,000 but is growing at 4000 people per year. When will both towns have equal population? Let’s Practice…

With a tailwind, an airplane makes a 900-mile trip in 2.25 hours. On the return trip, the plane flies against the wind and makes the trip in 3 hours. What is the plane’s speed? What is the wind’s speed? Let’s Practice…

Rate  Time = Distance

A chemist mixes a 20% saline solution and a 40% saline solution to get 60 milliliters of a 25% saline solution. How many milliliters of each saline solution should the chemist use in the mixture? Let t be the milliliters of 20% saline solution and f be the milliliters of 40% saline solution. Let’s Practice…

20% + 40% = 25% Saline

Suppose a pharmacist wants to get 30 g of an ointment that is 10% zinc oxide by mixing an ointment that is 9% zinc oxide with an ointment that is 15% zinc oxide. How many grams of each ointment should the pharmacist mix together? Let’s Practice…

Solving Systems of Inequalities by Graphing

Graphing Linear Inequalities

Details

Let’s Practice… Write an equation to represent the following:

Step 1: Re-arrange equations so y is alone, if needed. Step 2:Graph each equation a. Dotted or Solid? b. Shading? Step 3:Determine the point of intersection, if any. Step 4: Verify that the point of intersection determined in Step 3 is a solution of the system. Remember to check the point in both equations. Solving a System of Inequalities

Solve the system by graphing. Check your answer. Let’s Practice…

Solve the system by graphing. Check your answer. Let’s Practice…

Solve the system by graphing. Check your answer. Let’s Practice…

Solve the system by graphing. Check your answer. y > –2x + 5 y ≤ –2x – 4 Let’s Practice…

In one week, Ed can mow at most 9 times and rake at most 7 times. He charges $20 for mowing and $10 for raking. He needs to make more than $125 in one week. Show and describe all the possible combinations of mowing and raking that Ed can do to meet his goal. List two possible combinations. Let’s Practice…

Write the system of inequalities represented by the graph at the right. Let’s Practice…