3-1 Writing Linear Equations
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1 3-1 Writing Linear Equations Suppose you have a job working on a monthly salary of $2,000 plus commission at a car lot. Your commission is 5%. What would be your pay for selling the following in monthly sales? $15,360? $27,500? $53,900? $89,210? These could be written as ordered pairs and graphed. Do this below: (label axes and scale) 10K {(15,360, 2768), (27,500, 3375), (53,900, 4695), (89,210, )} Plot points but don t connect yet. 8K monthly pay 6K 4K 0 20K 40K 60K 80K 100K monthly sales Do you see that the relationship between sales and commission is linear? What does that mean? The points fall in a line when graphed. Would it make sense to connect these points and make it a line? _yes Why? _to include all the possible sales values (now connect with line) So a line is really a collection of ALL possible (x,y) values that pertain to this particular situation. If you squeezed in every single possible combination of sales and total pay, they would fit so tightly that you mine as well just draw a line... What points/characteristics would be special that we would want to pay attention to? Why? _steepness/slope, because it shows if it s increasing/decreasing and how fast y-intercept, because it shows the starting point, when sales (x value) are zero What about this graph? What other feature or special point might you be concerned about? _x-intercept/zero, because it shows where the inventory (y value) reaches zero inventory 2K car sales
2 What benefits would there be in representing linear situations such as these as graphs? _You can analyze the situation quickly and easily and project future values _ If we wanted to set a goal of a monthly income of $10,000, what would be an even better tool than this graph? linear equation/function for salary ( y = x ), because _you could plug in 10,000 and get an exact answer rather than just looking, and besides, this graph doesn t contain 10,000 on the y-axis Applications such as these are what makes writing, solving, and graphing linear equations so useful. The next few chapters we will be learning all about writing, solving, and graphing linear equations. Let s start with the ways you will see linear equations written. STANDARD FORM SLOPE-INTERCEPT FORM POINT-SLOPE FORM Ax + By = C y = mx + b (y-y₁) = m(x-x₁) Which of these forms was the above equation written in? slope-intercept But we could have rearranged the terms and written it in standard form. STANDARD Form Ax + By = C where:!! (x and y are variables),!! A, B, C are integers whose GCF is 1!! A and B are not both zero!! A > 0 (non-negative)
3 To write any linear equation in standard form: * Move all x and y terms to the left and constants to the right * Clear any decimals or fractions by multiplying by the LCM (fractions) or by multiplying by a multiple of 10 * Divide all the coefficients by their GCF. * If A is negative, multiply both sides by -1, because A must be positive. - If an equation cannot be written in this form, then it is not linear. - y = x Use the above steps to change into standard form: -.05x -.05x (100) -.05x + y = 2000 (100) -5x + 100y = 200,000 (gcf of 5, 100, is 5) 5 5 (-1) -x + 20y = 40,000 (-1) x - 20y = -40,000 Identifying Linear Equations Note that the following examples are not linear. Ex 1b 6x - xy =4 Ex 1B: y = x² - 4 Why not? _the xy term; it won t go in standard form (not linear) Why not? the x squared term; won t go in standard form; (not linear)_ So we can tell by looking at the equations that they are not linear. If we graphed these, they would have curves in them. We will graph them using calculators later in the lesson. (Tell them you won t have time but they can do it on their own later if they want.) Ex. 1A Is this linear? If so, write it in standard form. 1 3 y =!1 (3) (3) > y = -3 (do you see why it s in standard from? Look at checklist.) Ex. 1C Is this linear? If so, write it in standard form. 3 4 x = y + 8 A=0, B=1, C=-3 A, B, C are integers whose GCF is 1 A and B are not both zero A > 0 (non-negative) (move y left and multiply both sides by 4) > 3x - 4y = 32 So we can tell from the equations that these are linear. Let s graph the second one now and see.
4 Graphing by Making a Table Ex. 5 x y 3 4 x = y You can always graph any equation using a table of values. - Simply choose some x-values (be nice to yourself!) and solve for y (or vice versa). (Use wall grid board to graph) Always remember: * label your scale if anything other than 1 * use a straight edge and add arrows to the ends of the line * if graphing more than one line, label the lines with their respective equations Finding and Interpreting Intercepts Ex. 2 and 3 What is the x-intercept? 25 Name its ordered pair. (25,0) Interpret its meaning. _in 25 seconds, Trish s distance from the finish line was 0; she finished the race in 25 seconds What is the y-intercept? 200 Name its ordered pair. (0, 200)_ Interpret its meaning. _At time 0 seconds, she has 200 m to go; ie, she is at the starting line x-intercepts (-2,0 ) (3,0) none (-.5,0) y-intercepts (0,1) (0,2.5) (0,-3) (0,-.5) What do you notice about the y-coordinates of the x-intercepts? y-coordinate is zero What do you notice about the x-coordinates of the y-intercepts? x-coordinate is zero
5 Based on what you ve just discovered, if you want to find the intercepts of an equation (rather than a graph), what could you do? For x-intercept, set y=0 and solve for x; For y-intercept, set x=0 and solve for y. X-intercept (x,0) Y-intercept (0,y) Name the intercepts of y = 2x - 5. Write them as ordered pairs. x-int ( 2.5, 0) y-int (0, -5) One of the perks of equations in standard form is that they are very easy to graph using intercepts. Even though we used several values in the table to graph Ex. 1C, we only really needed how many? _2_ Since it only requires two points to graph any line, and since zeros are such easy numbers to work with, let s take the standard form of Ex. 1C that we found earlier and graph it by its intercepts. To graph by intercepts, start with an xy chart that looks like this: 0 0 Ex. 4 The standard form that we found in Ex. 1C was 3x - 4y = 32. Use this form to graph the line again, using intercepts. (Now graph using only intercepts. See how much 0-8 faster this is? If you want to check yourself, you can always 32/3 0 find another pair and make sure it s (10 and 2/3) on the line.) To check your graphing using your TI calculator, follow the steps on pages This is just to check. All graphing work must be done by hand, showing all work.
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