Systems of Linear Equations: Solve by Graph and Substitution

Systems of Linear Equations: Solve by Graph and Substitution

What is a system of equations?

• two or more equations solved simultaneously to see if they have a common solution

• can be the same type of function (as in both linear) or a combination of types of functions (as in linear and quadratic seen in the image)

Types of Solutions: No Solutions

• No solution: no intersection of the graphs.

• Systems with no solutions are called inconsistent.

• Parallel lines are an example of this type of system

• Linear systems with no solution will have the same slope and different y-intercepts

Types of Solutions: Infinite Solutions

• Infinition solutions: Lines represented differently but when expressed in slope-intercept form have the same slope and same intercept

• 4x + 3y = 10 and 8x + 6y = 20 are represented differently but both have a slope of -4/3 (-8/6 = -4/3) and a y-intercept of 10/3 (20/6 = 10/3)

• Called consistent and dependent

Types of Solutions: One solution

• Two lines that intersect in one point

• Will have different slopes

• Point of intersection is a solution to both equations

• Called Consistent and Independent

Multiple Choice

What is the solution?

• (6, 3)

• (3, 6)

• (-6, 3)

• No solution

Multiple Choice

What is the solution to this system?

• (3,1)

• No solution

• Infinite solutions

• (1,3)

Multiple Choice

How many solutions will this system have?

• No solution

• One Solution

• I Don't Know

• Infinitely Many Solutions

Multiple Choice

When you graph the exact same equation twice,

• you will have no solution.

• you will have one solution.

• you will have infinite solutions.

• you will graph a giraffe.

Multiple Choice

If a system of equations has no solution, what does the graph look like? 

• intersecting lines
• parallel lines
• skew lines
• intersecting lines

Multiple Choice

The solution of a system of linear equations is...

• the y-intercept (b)

• the intersection of the two equations on a graph

• the second equations' answers

• the slope (m)

Multiple Choice

What is the solution?

• One Solution

• No solution

• Infinitely Many Solutions

• (0, -1.5)

Ways to Solve: Graphing

• To solve a system of equations by graphing, graph each equation separately on the same coordinate plane

• If the equations are not in slope intercept form, put them into slope intercept form and graph.

• The point of intersection, if any, is the solution.

• The intersection (x,y) will be a solution to both equations.

Checking Your Solution

• Substitute your x-value and y-value from the point of intersection into each equation

• Evaluate

• If you get a true statement for each equation after your evaluate (3 =3, -5 = -5, for example) then the point is a solution

Multiple Choice

What is the solution?

• (1, -1)
• (-1, 1)
• (0, -2)
• (0, 1)

Multiple Choice

Buzz graphed two lines in order to find the solution to a given system of equations.
What is the solution?

• (-1, 4)
• (1, -4)
• (-4, 1)
• (4, -1)

Multiple Choice

Does the following system have One Solution, No Solution, or Infinite Solutions.

y = 4x + 8

y = -5x + 3

(Hint: Look at the slopes and y-intercepts. You don't have to graph)

• One solution

• No solution

• Infinite solution

Multiple Choice

Determine if (4, 1) is a solution for the system of equations.

y = -x + 5

y = 2x - 7

(Hint: Substitute the values for x and y in the equations. If it is a true statement, then the point is a solution)

• yes

• no

Fill in the Blanks

Fill in the Blanks

Multiple Choice

What is the solution?

• (6, 3)

• (3, 6)

• (-6, 3)

• No solution

Solving a System by Substitution: What is substitution?

• Substitute means to replace something with something that is equivalent

• Sugar can be replaced by a sugar substitute in a recipe; in math, we can replace an expression with something that has the same value

• In math, we replace equals with equals

Solving Linear Systems by Substitution

• Isolate a variable in one of the equations

• If 4x - 8y = 10, and you isolate x, that looks like 4x = 8y + 10; x = 8/4 y + 10/4; x = 2y + 5/2

• Substitute the expression for the variable in the second equation and solve for the variable

• Substitute the value for that value into one of the equations to find the other coordinate value of the solution

• If both equations are in slope intercept form, set the equations equal to each other and solve for x

• Then substitute the x-value in one of the equations to find the y-value

Multiple Choice

Solve the system by substitution. 
3x − 2y = 7 
y = 2x − 7 

• No solution 
•  (7, 7)  
•  (−7, −7)  
•  (−7, −3)  

Multiple Choice

When solving a system of equations algebraically, which statement results in infinite solutions?

• 4 = 7

• 2 = 2

• -3 = 3

• all of the above

Multiple Choice

Solve the system by substitution. 
y = 2x − 11 
2x − 8y = 4 

•  (6, 1)
• (6, −4)
• (6, −1)
• Infinite number of solutions  

Multiple Choice

Solve the system by substitution. 
-2x - 2y = 16
y = -8

• (0, -8)
• (-8, 0)
• (-16, -8)
• (-16, -8)

Applications of Linear Systems

Multiple Choice

On Monday Joe bought 10 cups of coffee and 5 doughnuts for his office at the cost of $16.50.  It turns out that the doughnuts were more popular than the coffee.  On Tuesday he bought 5 cups of coffee and 10 doughnuts for a total of $14.25.  Which equations could be used to determine the cost of the coffee? 

• 10c + 5d = 14.25
  5c + 10d = 16.50
• 10c + 5d = 16.50
  5c + 10d = 14.25
• c + d = 10
  5c + 10d = 16.50
• c + d = 5
  5c + 10d = 16.50

Multiple Choice

Some students want to order shirts with their school logo. One company charges $9.65 per shirt plus a setup fee of $43. Another company charges $8.40 per shirt plus a $58 fee. Which equation represents the number of shirts when both companies charge the same amount? 

• y = 9.65 + x
  y = 8.40 + x
• y = 9.65x + 43
  y = 8.40x + 58
• y =9.65x
  y = 8.40x
• y = 9.65x - 43
  y = 8.40x - 58

Multiple Choice

Alexandra finds that she can give 3 haircuts and 2 hair dyes in 315 minutes. Giving 2 haircuts and 4 hair dyes takes 450 minutes. Which system of equations represents the situation?

• 3x + 2y = 315

  2x + 4y = 450

• 3x + 2y = 450

  2x + 4y = 315

• 2x + 2y = 315

  3x + 4y = 450