3.1 Graphing Systems of Equations
A set of two or more equations that uses the same variables
What is a system of equations?
List the four steps required to solve a system of equations using substitution
1. Solve for one of the variables
2. Substitute the expression for y into the other equation and solve for x
3. Substitute the value of x into either equation and find the value for y
4. Write the solution in the form of an ordered pair
When graphing an inequality, the solutions are all of the ordered pairs either above or below this.
What is the boundary line?
The two methods you can use to solve a system of three equations with three variables.
What are substitution and elimination?
The term used for a solution of a system of two linear equations where the two lines cross
What is the point of intersection?
List the three steps required to solve a system of equations using elimination
1. Look for terms that are additive inverses
2. Substitute the value for x into one of the original equations to find the value for y
3. Write the solution in the form of an ordered pair
The graph shows that the boundary line is solid and shaded below it. What inequality would represent this scenario?
Less than or equal to
The terms for one of the variables in an equivalent system must have equal or opposite ___________ , so that when the equations are combined, the variable is eliminated.
What are coefficients?
The term used for determining the line of best fit mathematically
What is a linear regression?
Solve each system using substitution:
-4x + y = 8
2x - 3y = 6
(-3,-4)
You can find the solutions of an inequality system by graphing both inequalities on a coordinate plane and then doing what?
Finding the shaded area where they overlap
Describe the graph of a system of three equations with three variables that have no solution.
1. The graphs of the equations do not intersect
2. The graphs are parallel to each other
3. The graphs intersect in pairs
A system whose slopes and intercepts are different
What is an independent system?
Solve each system using elimination:
-2x + 6y = -10
x - 12y = 5
(5,0)
You are buying two kinds of notebooks for school. A three-ring notebook costs $6 and a spiral notebook costs $2. You must have at least five notebooks. The cost of the notebooks can be no more than $30. Write the two inequality statements that represent this real-world problem.
Inequality 1: number of three-ring notebooks + number of spiral notebooks ≥ 5
Inequality 2: cost of three-ring notebooks + cost of spiral notebooks ≤ 30
Describe the graph of a system of three equations with three variables that has one solution.
The graphs of the equations intersect at a single point
A system whose slopes are the same and intercepts are different
What is a dependent system?
Solving a system algebraically does not always result in a unique solution. You may get an equation with the same constants equal to each other, so the equation is always true. When this happens, the system is said to have what?
An infinite number of solutions
You are buying two kinds of notebooks for school. A three-ring notebook costs $6 and a spiral notebook costs $2. You must have at least five notebooks. The cost of the notebooks can be no more than $30.
Let x = the number of three-ring notebooks.
Let y = the number of spiral notebooks.
Let 6x = the cost of three-ring notebooks.
Let 2y = the cost of spiral notebooks.
Graph the solution to the above systems
The graphed solution of:
x + y > or equal to 5
6x + 2y < or equal to 30
Solve each system using the method of your choice:
3x - 2y - 5z = -10
-x - 2y + z = 10
x + 2y + 5z = 2
(-2, -3, 2)