Section 3.1: Graph Linear Equations in Two Variables

Section 3.1: Graph Linear Equations in Two Variables Chapter 3: Lines This section will focus on graphing lines. Here are the forms of the most common equations that graph into lines. In this section we will focus on the general form and vertical and horizontal lines. How to create a graph of a Linear Equation written in General Form: 𝐴𝑥 + 𝐵𝑦 = 𝐶 by plotting points. o o Every point on the line is a solution of the equation. Every solution of this equation is a point on this line. Step 1. Pick 3 values to substitute into 𝑥 in the given equation and solve for 𝑦. This will create 3 points to plot Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work. Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of line. For Example: Graph: 3𝑥 + 6𝑦 = 12 Step 1. Pick three values for x and solve for y. This is the step that can be tricky. You may pick any 3 values for x. If you choose wisely you will avoid fractions. If your choices are not carefully selected, you will have to deal with fractions. I will try to explain how I select the values to avoid fractions during class as it is quite a task to type my thought process. I will choose: 𝑥 = 0,2 𝑎𝑛𝑑 4 (You literally can pick any 3 numbers for x. If you pick a number and the algebra gets messy, throw it out and try another value.) Step 1: x=0 3(0) + 6𝑦 = 12 6𝑦 = 12 𝑦=2 Point (0,2) Step 2: Plot the 3 points x=2 3(2) + 6𝑦 = 12 6 + 6𝑦 = 12 6𝑦 = 6 𝑦=1 Point (2,1) x=4 3(4) + 6𝑦 = 12 12 + 6𝑦 = 12 6𝑦 = 0 𝑦=0 Point (4,0) Step 3 connect the points with a line Section 3.1: Graph Linear Equations in Two Variables Chapter 3: Lines x-intercept and y-intercept of a Line • The x-intercept is the point (𝑎, 0) where the line crosses the x-axis. The x-intercept occurs when 𝑦 = 0. • The y-intercept is the point (0, 𝑏) where the line crosses the y-axis. The y-intercept occurs when 𝑥 = 0. How to find the 𝑥 − intercept and 𝑦 − intercept from the equation of a line and then use them to sketch a graph of the line. Step 1. Find the x- and y-intercepts of the line. Let 𝑦 = 0 and solve for x to find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. Let 𝑥 = 0 and solve for y to find the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. Step 2. Find a third solution to the equation by picking any number and plugging it in for x solve for y. Step 3. Plot the three points and check that they line up. Step 4. Draw the line For Example: Find the x-intercept and y-intercept from the equation of a line and sketch a graph of the line. 2𝑥 − 4𝑦 = 8 Step 1: Find 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 by letting 𝑥 = 0. Find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 by letting 𝑦 = 0. 2(0) − 4𝑦 = 8 2𝑥 − 4(0) = 8 −4𝑦 = 8 2𝑥 = 8 𝑥=4 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (4, 0) 𝑦 = −2 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (0, −2) Step 2: Pick a value for x and use it to find another point. I will pick 𝑥 = 2. 2(2) − 4𝑦 = 8 4 − 4𝑦 = 8 −4𝑦 = 4 𝑦 = −1 𝑒𝑥𝑡𝑟𝑎 𝑝𝑜𝑖𝑛𝑡 (2, −1) Section 3.1: Graph Linear Equations in Two Variables Step 3: Plot the three points: (0, −2) (4,0) 𝑎𝑛𝑑 (2, −1) Chapter 3: Lines Step 4: Connect the points with a line. Graphs of vertical lines in the form 𝑥 = 𝑎 The graph of an equation of the form 𝑥 = 𝑎 is a vertical line. The line passes through the 𝑥 − 𝑎𝑥𝑖𝑠 at (𝑎, 0). Find three points and graph the vertical line 𝑥 = 4. Step 1: Find three points. Since the equation I am asked to graph has only I variable I am going to find my points using logic as opposed to using Algebra. • The equation only as an x. Every point on the graph must meet the requirement that 𝑥 = 4. • The equation does not have a 𝑦. This implies that 𝑦 can assume any numerical value. • Any point of the form (4, #) will meet the only requirement: 𝑥 = 4 • Create any points that have a 4 as the x-coordinate and they will be on the graph of 𝑥 = 4. The points I choose are (4,0) (4,-2) and (4,3). Really, any three points that have 4 for x will work. Step 2: Plot the three points Step 3: Connect the points with a line. Section 3.1: Graph Linear Equations in Two Variables Chapter 3: Lines Graphs of horizontal lines in the form 𝑦 = 𝑏 The graph of an equation of the form 𝑦 = 𝑏 is a horizontal line. The line passes through the 𝑦 − 𝑎𝑥𝑖𝑠 𝑎𝑡 (0, 𝑏). Find three points and graph the vertical line 𝑦 = 6. Step 1: Find three points. Since the equation I am asked to graph has only I variable I am going to find my points using logic as opposed to using Algebra. • The equation only as an y. Every point on the graph must meet the requirement that y = 6. • The equation does not have a x. This implies that x can assume any numerical value. • Any point of the form (#, 6) will meet the only requirement: 𝑦 = 6 • Create any points that have a 6 as the y-coordinate and they will be on the graph of y = 6. The points I choose are (0,6) (-5,6) and (3,6). Really, any three points that have 6 for y will work. Step 2: Plot the three points Step 3: Connect the points with a line.