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A linear equation in two variables: Is of the form Ax + By = C A, B, C are real numbers A & B are not both zero. (It’s OK for one or the other to be zero, just not both at the same time.) This is called the “standard form” of a linear equation. “Standard form” has the x and y terms on the left and the constant (number) term on right, and usually, all fractions cleared away by multiplying by LCD. (We’ll also be studying two other forms of a linear equation: “slope-intercept” form and “point-slope” form.)

Example: Graph the linear equation 2x - y = -4. Process: Find two ordered pair solutions (and a third solution as a check of the computations) by choosing a value for one of the variables, x or y, then solving for the other variable. We plot the solution points, then draw the line containing the 3 points. Graphing linear equations: Find at least 2 points on the line. Connect the points to form a line.

Example (cont.) Graph the linear equation 2x - y = -4. ( NOTE: You can pick any number you want for x, so pick something simple and easy, like 0 or 1 or 2 or -1 or -2.) Let x = 1. Then 2x - y = -4 becomes 2(1) – y = -4 (replace x with 1) 2 – y = -4 (simplify the left side) -y = -4 – 2 = -6 (subtract 2 from both sides) y = 6 (multiply both sides by -1) So one solution is (1, 6).

Example (cont.) Graph the linear equation 2x - y = -4. For the second solution, let y = 4. (Could be any number you choose.) Then 2x - y = -4 becomes 2x – 4 = -4 (replace y with 4) 2x = (add 4 to both sides) 2x = 0 (simplify the right side) x = 0 (divide both sides by 2) A second solution is (0, 4).

Example (cont.) Graph the linear equation 2x - y = -4. For the third solution, let x = -3. (Could be any number you choose.) Then 2x - y = -4 becomes 2(-3) – y = -4 (replace x with -3) -6 – y = -4 (simplify the left side) -y = = 2 (add 6 to both sides) y = -2 (multiply both sides by -1) A third solution is (-3, -2). (Note: different people would probably pick different numbers and get three different points, and they’d still come out with the same graphed line.)

Now we plot all three of the solutions (1, 6), (0, 4) and (-3, -2). x y (1, 6) (0, 4) (-3, -2) And then we draw the line that contains the three points.

Helpful Hint When graphing a linear equation in two variables: if it is solved for y, it may be easier to find ordered pair solutions by choosing x- values. If it is solved for x, it may be easier to find ordered pair solutions by choosing y- values.

Example Graph the linear equation y = x + 3. Since the equation is solved for y, we should choose values for x. To avoid fractions, we should select values of x that are multiples of 4 (the denominator of the fraction). (The online graphing tool usually does not allow you to graph fraction coordinates, so you need to select values of x and y that will make both coordinates integers.)

Example (cont.) Let x = 4. y = = 6 (simplify the right side) One solution is (4, 6). Graph the linear equation y = x + 3. Then y = x + 3 becomes y = (4) + 3 (replace x with 4)

Example (cont.) For the second solution, let x = 0. (Zero is always a good, easy-to-graph number to pick for one of your values.) y = = 3 (simplify the right side) A second solution is (0, 3). Graph the linear equation y = x + 3. Then y = x + 3 becomes y = (0) + 3 (replace x with 0)

Example (cont.) For the third solution, let x = -4. y = = 0 (simplify the right side) A third solution is (-4, 0). Graph the linear equation y = x + 3. Then y = x + 3 becomes y = (-4) + 3 (replace x with -4)

Now we plot all three of the solutions (4, 6), (0, 3) and (-4, 0). x y (4, 6) (0, 3) (-4, 0) And then we draw the line that contains the three points.

Recall the definition of a linear equation in two variables: It is of the form Ax + By = C A, B, C are real numbers A & B are not both zero. (It’s OK for one or the other to be zero, just not both at the same time.) Question: Can we classify an equation like x = 5 or y = -11 as a “linear equation in two variables”? So back to the question: Can we call x = 5 and y = -11 “linear equation in two variables”? What would their graphs look like in 2 variables?

xy Example: Graph the linear equation y = 3 on a two- variable graph grid. Notice that the equation can be written in standard form as 0x + y = 3. No matter what value we replace x with, y is always 3.

Besides linear equations, we will also examine the graphs of quadratic equations, absolute value equations and other nonlinear equations in later sections. Non-linear simply means the graph is not a straight line.

One type of a non-linear equation is the quadratic. A quadratic has ONE of the variables squared (we’ll use x as the squared variable for examples in this section). The graph of a quadratic equation is shaped like a “U” with the ends either pointing upwards (like a cup) or downwards (like a hill.) To graph a quadratic equation, plug values in the equation for the x and calculate the corresponding y values to get the ordered pairs to plot as the points on your graph. You might need a LOT of points – go until you find where the graph reaches a turning point and starts curving back in the other direction.

x y Example Graph y = 2x 2 – 4. x y (2, 4) (-2, 4) (1, -2)(-1, -2) (0, -4)

Application problem from today’s homework that can be solved by graphing: The coordinates of three vertices of a rectangle are (7, -4), (9, -4), and (7, 7). Find the coordinates of the fourth vertex.

The assignment on this material (HW 3.2) Is due at the start of the next class session. Visit the MathTLC for homework help! Please remember to sign in on the Math 110 clipboard by the front door of the lab

You may now OPEN your LAPTOPS and begin working on the homework assignment. We expect all students to stay in the classroom to work on your homework till the end of the 55- minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice test.