3.1 Testing H5P B

Learning Objectives

After completing this section, you should be able to:
  1. Plot points in a rectangular coordinate system.
  2. Verify solutions to an equation in two variables.
  3. Complete a table of solutions to a linear equation.
  4. Find solutions to a linear equation in two variables.

Introduction: The Coordinate Plane

Many real-world relationships involve two quantities that change together, such as time and distance, or price and quantity.
To study these relationships visually, we use the rectangular coordinate system (or Cartesian plane).
The plane is formed by two perpendicular number lines:
  • The horizontal axis is called the x-axis.
  • The vertical axis is called the y-axis.
Their intersection is the origin (0,0).

The Four Quadrants

The axes divide the plane into four quadrants:
Cartesian Grid labeled into its 4 quadrants
Figure 3.1A
Table 3.1A
Quadrant x-sign y-sign
I + +
II – +
III – –
IV + –

Mental Shortcut

Quadrant I is (+,+). Moving counterclockwise: Quadrant II has negative x, Quadrant III has both negative, and Quadrant IV has negative y.

Plotting Points

Each point in the plane is identified by an ordered pair (x,y).
  • The x-coordinate shows the horizontal position.
  • The y-coordinate shows the vertical position.
Together, they locate the point relative to the origin.

Example 3.1.1 — Plot and Identify Quadrants

Plot the following points and identify which quadrant each lies in:
A(3,5),\; B(-2,4),\; C(0,4),\; D(-3,0),\; E(-2,-3),\; F(3,-3),\; G(4,-1)
Blank Grid x = -3 to 4; y =-4 to 5
Figure 3.1B
Solution
• A(3,5) → Quadrant I
• B(-2,4) → Quadrant II
• E(-2,-3) → Quadrant III
• F(3,-3) → Quadrant IV
• C(0,4) → on the y-axis
• D(-3,0) → on the x-axis
• G(4,-1) → Quadrant IV
3_1AnswerGrid
Figure 3.1C

Try It

Try It

 

Naming Coordinates from a Graph

Sometimes the coordinates are not given—you must read them from a grid.

Example 3.1.2 — Read Ordered Pairs from a Graph

From the graph, name the ordered pair for each labeled point A–G.
Grid with labeled Points
Figure 3.1D
Solution
The first coordinate is x and the second coordinate is y:
A(-1, 1); B(-2, -4); C(0, -4); D(4, 0); E(-2, 3); F(2, 3); G(4, -2)

Try It

Now that we can plot and read points, we can connect points to equations: a point is on the graph exactly when it makes the equation true.

Verifying Solutions to an Equation in Two Variables

An equation in two variables (such as y = 2x + 1 or 3x + 2y = 12) has infinitely many ordered pairs that make it true.
Each true ordered pair is a solution, and every solution corresponds to a point on the line representing that equation.

Example 3.1.3 — Check Ordered Pairs

Determine which of the following ordered pairs are solutions to 2x + y = 6.
A(1,4),\; B(3,0),\; C(2,3)
Solution
Substitute each ordered pair into the equation:
• (1,4):\; 2(1) + 4 = 6 ✓
• (3,0):\; 2(3) + 0 = 6 ✓
• (2,3):\; 2(2) + 3 = 7 ✗
Solutions: (1,4) and (3,0)

Try It

Completing a Table of Solutions

To graph a linear equation, it helps to find several solutions and record them in a table.
Tip: Choose x-values that make the arithmetic easy and the points easy to plot.

Example 3.1.4 — Build a Table

Find three points that satisfy y = 2x - 4.
Solution
Choose convenient x-values: 0, 1, and 2. Substitute to find the corresponding y-values.
Table 3.1B
x y Ordered Pair
0 -4 (0,-4)
1 -2 (1,-2)
2 0 (2,0)
These three points lie on the same straight line.

Line though the points (0,-4), (1,-2), (2,0).
Figure 3.1E

Try It

Finding Missing Coordinates

You can find missing x- or y-values by substituting into the equation and solving for the unknown.

Example 3.1.5 — Find Missing Coordinates

For the line y = -x + 5:
a) If x = 4, find y.
Solution
y = -4 + 5 = 1
So the ordered pair is (4,1).
b) If y = 2, find x.
Solution
2 = -x + 5
x = 3
So the ordered pair is (3,2).
These can also be displayed together in a table.
Table 3.1C
x y
4 1
3 2

Graphing from a Table

Once you have at least two points, you can graph the line representing all solutions.
Key idea: If you plot two or more solutions and draw a straight line through them, that line is the graph of the equation.

Example 3.1.6 — Graph a Linear Equation

Graph y = -2x + 8.
Solution
Create an x–y table with convenient x-values:
Table 3.1D
x y Ordered Pair
0 8 (0,8)
1 6 (1,6)
2 4 (2,4)
Plot the three points and draw a straight line through them.
Points (0,8), (1,6), (2,4) with a line through them
Figure 3.1F
This line represents every ordered pair that satisfies the equation.

Example 3.1.7 — Graph a Linear Equation with a Fractional Slope

Graph y = \frac{3}{4}x - 1.
Solution
Because the slope is a fraction, choose x-values that are multiples of 4 to avoid fractions in the table:
Table 3.1E
x y Ordered Pair
0 -1 (0,-1)
4 2 (4,2)
8 5 (8,5)
Plot the three points and draw a straight line through them.
Line through the points (0,-1), (4,2) and (8,5)
Figure 3.1G
This line represents every ordered pair that satisfies the equation.

Try It

While difficult to graph on the computer, you should be able to create a grid like the one below (yours may need to be scaled differently) and transfer the points to the grid and graph.
5x5 Blank grid
Figure 3.1H

Quick Check

• Which axis is (0,4) on? → y-axis
• Which coordinate comes first in (x,y)? → x
• What quadrant is (-4,5)? → Quadrant II
• What point is the origin? → (0,0)
• What must be true in Quadrant IV? → x>0,\; y<0

Key Takeaways

  1. The coordinate plane represents all solutions to an equation visually.
  2. Ordered pairs give both direction and distance from the origin.
  3. To verify a solution, substitute both coordinates into the equation.
  4. Linear equations have infinitely many solutions that lie on a straight line.
  5. Tables and graphs connect algebraic and graphical thinking.

 

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