Solve The Systems by Substitution Method

 Key Concepts

• Use substitution to solve a system of equations with one solution
• Use substitution to solve a system with no solution
• Use substitution to solve a system with infinitely many solutions

Solve Systems by Substitution

1. Identify the slope and y-intercept of the equation y= – 0.5x +2.5 are ………. slope and  
   y-intercept……. 

2. A …………………………. is a relationship between two variables that gives a straight line when graphed. 

3. y – x =5 Solve the equation for y ……….. 

4. y = x + 4 and y = –x + 6. How many solutions for this system of equations have?…………. 

5. If a system has no solution, what do you know about the lines being graphed? 

6. What value of ‘m’ gives the system infinitely many solutions? –x + 4y = 32 and  
   y= mx + 8. ………… 

7. How many solutions does the system have? See the graph. 

8. Can you solve the system of equations without slope-intercept form or graphing? 

Answers: 

1. Slope =–0.5 and y-intercept=2.5 

2. Linear equation 

3. y=x+5 

4. One 

5. Parallel lines 

6. 1414 

7. One solution 

8. I don’t know  

Let’s learn about a new topic to solve the system of equations. 

Explain It! 

Jackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rate for on Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On time Cabs will cost less. 

1. Do you agree with Jackson? Explain. 

Solution: 

No. 

Destination from Jackson =4 miles 

Slope of On Time Cabs = m₁

Slope of Speedy Cabs = m₂

Slope of On Time Cabs <  Slope of Speedy Cabs 

 m₁< m₂

By observing the graph, the solution is (4, 10) 

From Jackson’s destination the cost is equal for both cabs, i.e., $10 per cab. 

So, the cost is not dependent on the slope. 

2. Which taxi service company should Jackson call? Explain your reasoning. 

       Jackson can choose either Speedy cabs or On time cabs. 

The cost is the same for both cabs. 

Focus on math practices 

Be Precise. Can you use the graph to determine the exact number of miles for which the cost of the taxi ride will be the same? 

Answer: 

Sometimes the graph cannot determine the exact number of miles for which the cost of the taxi ride will be the same. 

So, we can choose another way to solve the system of equations.  

Ok, let’s know about that, 

Systems of Equations: A set of equations is called a system of equations. The solutions must satisfy each equation in the system. 

Systems of Linear Equations:  

A solution to a system of equations is an ordered pair that satisfy all the equations in the system.  

A system of linear equations can have:  

1. Exactly one solution  

2. No solutions  

3. Infinitely many solutions 

Systems of Linear Equations:  

There are four ways to solve systems of linear equations: 

1. By slope and intercept form  

2. By graphing  

3. By substitution  

4. By elimination  

SOLVE A SYSTEM OF EQUATIONS BY SUBSTITUTION. 

Steps: 

1. Solve one of the equations for either variable.  

2. Substitute the expression from Step 1 into the other equation.  

3. Solve the resulting equation.  

4. Substitute the solution in Step 3 into one of the original equations to find the other variable.  

5. Write the solution as an ordered pair.  

6. Check that the ordered pair is a solution to both original equations. 

Use Substitution to Solve a System of Equations with One Solution: 

Example 1: 

Gemma sells tickets at a local fair. On Saturday, she sells 800 tickets and collects a total of $7,680. How can Gemma determine the number of each type of ticket that was sold on Saturday? 

Solution: 

Step 1:  

Write a system of linear equations to represent the situation. 

Children’s tickets + Adult Tickets = Total tickets 

               C +            a = 800  

           7.50c +      13.50a = 7,680 

Amount Collected + Amount Collected = Total Amount 

For Children for Adult Collected  

Tickets                        Tickets 

Step 2: Solve for one of the variables. 

First, solve one of the equations for one variable. 

   c + a = 800       ………………You can solve for a or c  

          c  = 800 – a 

Substitute 800 – a for c in the other equation, then solve. 

7.50 (800 – a) + 13.50a = 7,680 

7.50 × 800 – 7.50 × a +13.50a = 7680 ……..Distributive property across subtraction 

6,000 – 7.50a + 13.50a = 7,680 

            6a  = 1,680 

              ∴                  a = 280 

∴ 280 adult tickets were sold 

Step 3: Solve for the other variable. 

                c + a = 800 

     c   + (280) = 800  

               –280       –280………… Subtract – 280 both sides  

                      c   = 520 

∴ 520 children’s tickets and 280 adult tickets were sold on Saturday. 

Example 2: 

Seiko takes pottery classes at two different studios. Is there a number of hours for which Seiko’s cost is the same at both studios? 

Solution: 

Step 1:  

Write a system of linear equations to represent the situation. Let x equal the number of hours and y equal Seiko’s cos 

              y = 14x              

              y = 1/2 (28x + 15) 

Step 2:  

Use substitution to solve one of the equations for one variable. 

The result is not a true statement, so the system has no solution. There is no number of hours for which Seiko’s is the same at both studios. 

Example 3: Solve the system –2y – 3x = –84.91 and 3x + 6y = 254.73 by using Substitution. 

Solution: 

Step 1:   

Solve one of the equations for one variable. 

          -2y – x = -84. 91 

                 – x = -84.91 + 2y 

                 ∴x = 84.91 – 2y            …….. multiply both sides with –1 

Reasoning  

You can use either equation to solve for either variable. How do you decide which equation and which variable to choose? 

Step 2:  

Substitute 84.91 – 2y for x in the other equation. 

Then solve. 

Solve Systems by Substitution method: 

Problems and solving. 

1. Solve each system by substitution method. 

Describe the solutions. 15x+y=2 and 10x+y=3. 

Solution: 

Step 1:   

Given equations are 15x+y=2 and 10x+y=3. 

Solve one of the equations for one variable. 

               10x + y = 3 

                            y = 3 – 10x 

Step 2: Substitute 3 – 10x for y in the other equation 10x + y =3. 

               Then solve. 

Step 3: 

Now substitute the x value in the y equation.  

Let’s Check your Knowledge: 

1. Solve the system by substitution method 2x – y = 6   and   6x + 2y = 4  

2. Solve the system by substitution method  

  3x + 4y = 33 

  2x + y = 17  

3. Solve the system by substitution. 

– x + y =3 and x + y = 6 

Answers: 

1. Solve the system by substitution method 2x – y = 6 and 6x + 2y = 4 

Solution:  

Now substitute the x value in the other equation. 

                y = 2x – 6  

                y = 2 (1.6) – 6 

                y = 3.2 – 6  

                y = –2.8  

Solution for system is x = 1.6 and y = –2.8 

The system has one solution. 

Key concept: 

Systems of linear equations can be solved algebraically. When one of the equations can be easily solved for one of the variables, you can use substitutions to solve the system efficiently. 

Step 1: Solve one of the equations for one of the variables. 

Then substitute the expression into the other equation and solve. 

Step 2: Solve for the other variable using either equation. 

Exercise:

1. Remo’s age is three times the sum of the ages of his two sons. After 5 years, his age will
   be twice the sum of the ages of his two sons. Find the age of Remo.
2. The middle digit of a number between 100 and 1000 is zero, and the sum of the other
   digit is 13. If the digits are reversed, the number so formed exceeds the original number
   by 495. Find the number.
3. The sum of the present ages of Max and Ranjith is 11 years. Max is older than Ranjith by
   9 years. Find their present ages.
4. If 1 is added to each of the given two numbers, then their ratio is 1:2. If 5 is subtracted
   from each of the two numbers, then their ratio is 5:11. Find the numbers.
5. A piece of work is done by 6 men and 5 women in 6 days or 3 men and 4 women in 10
   days. How many days will it take for 9 men and 15 women to finish that work?

Concept Map: