5.1 – Two-Variable Systems: Graphing

Key Terms

• Axes – The reference lines used in a graph. (Axes is the plural of axis.)
• The axes of a coordinate graph are perpendicular.
• The x-axis is horizontal, and the y-axis is vertical.
• Intersect – To cross over one another.
• System of Two Linear Equations – Groups of two linear equations that have the same variables and are used together to solve a problem.
• A linear equation can be written in the form y = mx + b.
• The graph of a linear equation is a straight line.

Review

• Coordinates are alphabetical.  x comes first, then y: (x, y)
• On a graph, you plot x first (horizontally), then y (vertically).
• You mark this point with a dot and name the coordinates with their respective numbers for (x, y).
• Ex.  (2, -3) is located on the graph below.

Notes

• Solutions Sets
• One Solution – when multiple lines cross at one single point
• They have unique slopes and y-intercepts.

• Zero Solutions – when multiple lines never cross (they are parallel)
• They have the same slope, but different y-intercepts.

• Infinitely Many Solutions – when multiple lines lie on top of one another and share all of the same points
• They have the same equation; so, same slope and same y-intercept.

Examples

• Ex 1. What is the solution to the system of equations graphed below?
• Line 1:  y = 2x – 7
• Line 2:  y = -x – 1
• It’s easy to see the answer by tracing the x and y coordinate to the point where the lines meet.  They meet at (2, -3).
• However, there is another way to solve for the solution point if you can’t trace it on a graph:
• Since y is equal to both “2x – 7” and also “-x – 1,” you can set both equations equal to one another:
• Setup: 2x – 7 = -x – 1
• Add 7 to both sides: 2x = -x + 6
• Add x to both sides: 3x = 6
• Divide both sides by 3: x = 2
• Substitute x = 2 into either of the two equations.  Let’s pick y = 2x – 7.
• y = 2 (2) – 7
• Simplify: y = 4 – 7; so, y = -3
• So, x = 2 and y = -3
• Answer in coordinate form: (2, -3)

• Ex 2. The owner of a bike shop sells unicycles and bicycles and keeps inventory by counting seats and wheels.
• One day, she counts 20 seats and 28 wheels.
• a. Which equation in slope-intercept form shows that the number of bicycles (x) plus the number of unicycles (y) is 20?
• Setup: x: bicycles and y: unicycles
• The total is 20, so x + y = 20.
• Rearrange to slope-intercept form by subtracting x on both sides.
• Answer: y = -x + 20
• b. Which equation in slope-intercept form shows that x bicycles and y unicycles together have a total of 28 wheels?
• Setup: x: bicycles and y: unicycles
• 2x would represent the wheels on a bicycle
• y would represent the wheels on a unicycle
• The total wheel count is 28; so, 2x + y = 28.
• Rearrange to slope-intercept form by subtracting 2x on both sides.
• Answer: y = -2x + 28