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 Blank

What is the solution?

** Write it as an ordered pair such as: (-1, 2)

Fill in the Blank

What is the solution?

** Write it as an ordered pair such as: (-1, 2)

Fill in the Blank

If a system of equations have two lines that overlap, meaning that they share all points, how many solutions does the system have?

Fill in the Blank

If two lines intersect ONCE, how many solutions will the system of equations have?

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