STRAIGHT LINES 1



STRAIGHT LINES 1 MTH-3003-2 (Mathematics Preparation)

Unit 1 Representing Points in the Cartesian Plane

The x-axis and the y-axis of the Cartesian plane represent two number lines, one horizontal and one vertical. The abscissa is usually represented by the letter x and the ordinate by the letter y. The two coordinates of the point therefore give the ordered pair (x , y). The dependent variable is represented by y and the independent variable by x.

To Locate a Point in the Cartesian Plane:

1. Locate the value of the abscissa on the x-axis.

2. Draw a broken line parallel to the y-axis.

3. Locate the value of the ordinate on the y-axis.

4. Draw a broken line parallel to the x-axis.

5. Mark a dot at the intersection of these broken lines.

6. Label the coordinates of the point.

To determine the coordinates of a point:

1. Starting from the given point, draw a broken line parallel to the y-axis and identify the abscissa.

2. Starting from the given point, draw a broken line parallel to the x-axis and identify the ordinate.

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Scale : x-axis: 1 division = 1 unit y-axis: 1 division = 1 unit

Unit 2 Graphing a Linear Equation given a table of Values

Standard linear equation : y = m x + b

To tabulate the standard linear equation, given a constant value for m and b, assign different values to the variable x and calculate the respective values of y. Scale the x and y axes accordingly. Identify the collinear points and draw a straight line through them.

Any equation in which the variables are of the first degree is called an equation of the first degree.

An equation of the first degree in two variables is always represented graphically by a line. It is called an equation of a line or a linear equation.

To complete a table of values and to draw the line corresponding to an equation:

1. Assign any numerical value to one of the two variables.

2. Calculate the value of the other variable.

3. Repeat this operation for a minimum of four (4) ordered pairs (x, y).

4. Plot the points obtained in a Cartesian plane.

5. Check to see whether the points are collinear.

6. Draw the line.

Example : Graph the linear equation representing an electricity bill where the amount due is $0.04 per kilowatt- hour (kWh) used plus a fixed charge of $10.00. The required mathematical formula is

Amount due = ($0.04 × kWh) + $10.

Let the x-axis = kWh and the y-axis = the amount due, we have the equation y = 0.04x + 10 to graph.

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Number of kWh consumed

Unit 3 Determining Points given the Graph of a Linear Equation:

To know which measurement unit to use, you must consider the maximum possible value of the variable to be located on the axis and the size you want to make your Cartesian plane, that is, the space you will have in which to draw your graph.

To graph a line and to plot other points in a Cartesian plane:

1. Label the axes.

2. Graduate the axes, using an appropriate unit of measure for the situation (scale).

3. Plot the two given points in the Cartesian plane.

4. Draw a line through these points.

5. Plot the other points by determining the value of x or y as the may be case.

Example : Ms. Poirot sells refrigerators. His salary varies according to the number of refrigerators he sells. Three weeks ago he sold 15 refrigerators for a salary of $250. During a special promotion, he was able to sell triple this number, that is 45 refrigerators, and he earned $550. If Ms. Poirot sold 25 refrigerators, what his salary be ? How many refrigerators must he sell to receive $400 ?

His salary depends on the number of refrigerators sold, therefore the dependent variable y represents his salary and the independent variable x represents the number sold.

The maximum number of refrigerators being considered is 45, therefore we must arrange our x-axis scale for this number. The maximum salary considered is $550, therefore we must arrange our y-axis scale for this number.

The ordered pair to graph is (x , y) = (# of fridges , salary).

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Number of refrigerators sold

Unit 4 Graphing an Equation of the Form Ax + By + C = 0

The x-intercept is the value of the variable x where the line intersects the x-axis, that is, the value of x where y is zero (0).

The y-intercept is the value of the variable y where the line intersects the y-axis, that is, the value of y where x is zero (0).

To graph an equation of the form Ax + By + C + 0 where A, B ( 0:

1. Isolate the variable y, if necessary.

2. Determine the x-intercept and the y-intercept to resolve the ordered pairs (x , 0) and (y , 0).

3. Determine three (3) other ordered pairs that satisfy the equation.

4. Record these ordered pairs in a table of values.

5. Graduate the axes in the Cartesian plane.

6. Plot the ordered pairs obtained, in the Cartesian plane.

7. Draw a solid line through these points.

8. Identify the values of the intercepts in this Cartesian plane.

Note, when isolating a variable, always change to a positive variable. Thus if (y = 2x ( 3, change the signs on both sides of the equation to obtain y = (2x + 3.

All equations of the form By + C = 0, that is, equations where the x variable does not appear, are represented graphically by a horizontal line parallel to the x-axis.

To graph an equation of the form By + C = 0:

1. Isolate the variable y.

2. Determine three (3) to five (5) ordered pairs that satisfy the equation.

3. Record these ordered pairs in a table of values.

4. Graduate the axes of the Cartesian plane.

5. Plot the ordered pairs obtained, in the Cartesian plane.

6. Draw a horizontal line through these points.

7. Identify the coordinates of the points in the line.

All equations of the form Ax + C = 0, that is, equations where the y variable does not appear, are represented graphically by a vertical line parallel to the y-axis.

To graph an equation of the form Ax + C = 0:

1. Isolate the variable x.

2. Determine three (3) to five (5) ordered pairs that satisfy the equation.

3. Record these ordered pairs in a table of values.

4. Graduate the axes of the Cartesian plane.

5. Plot the ordered pairs obtained, in the Cartesian plane.

6. Draw a vertical line through these points.

7. Identify the coordinates of the points in the line.

Example : Graph the equation 4x + 6y – 12 = 0

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Example 2 : Graph the equations (a) 3y – 6 = 0 and (b) [pic]

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Unit 5 Determining the Slope (Rate of Change) of a Line given Two points

The inclination (mathematical slope) is the ratio of the height (h) to the base (b) of the right triangle by drawing a horizontal line from the start of the slope to meet the vertical line drawn from the highest point of the slope, namely the ratio h ( b (rise ( run).

The higher the ratio of the height to the base of the right triangle, the greater the slope.

The slope (inclination) of a horizontal line is zero.

To calculate the slope of a line from a graph illustrating a situation :

1. Draw a right triangle by drawing a broken horizontal line parallel to the x − axis and a vertical broken line parallel to the y – axis, starting at the two reference points.

2. Calculate the vertical change or rise (the height of the triangle) and the horizontal change or run (the base of the triangle) and determine the sign of each change.

3. Determine the slope of this line by establishing the ratio of the vertical change to the horizontal change.

4. The slope or rate of change is the ratio of the vertical change in a variable to the horizontal change in the other variable. That is

[pic]

Note: Percent is a ratio whose denominator has been converted to 100. Thus [pic].

A fraction is converted to a decimal by dividing the numerator by the denominator.

Thus [pic]becomes 1 ( 4, that is 0.25.

The unit of measure of a slope is determined by the unit of measure of the variable located on the y-axis with respect to the unit of measure of the variable located on the x-axis.

Whenever the signs of the vertical and horizontal displacements are different, the slope (rate of change) of the line is negative.

Whenever the signs of the vertical and horizontal displacements are the same, the slope (rate of change) of the line is positive.

The greater the slope (rate of change), the steeper or the closer to the vertical is the line.

The slope of a horizontal line, a line parallel to the x-axis, is always zero.

A displacement is positive when it is effected in the increasing direction of the axes, namely, upward with respect to the y – axis or to the right with respect to the x – axis

A displacement is negative when it is effected in the decreasing direction of the axes, namely, downward with respect to the y – axis or toward the left with respect to the x – axis

Example : The graph below shows the distance traveled by a cyclist, in relation to the time elapsed. The information recorded on the graph determines a straight line. Using two of thee points, calculate the slope or rate of change of this line, that is, the rate of change of the distance traveled in relation to the time elapsed.

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Example : Calculate the slope (rate of change) of the savings, illustrated in the graph below, in relation to the time in months. Considering the two ordered pairs shown, (a) What is the length and direction of the vertical displacement? (b) What is the length and distance of the horizontal displacement? (c) What is the slope of the line in the appropriate units of measure? (d) What is the sign of the slope?

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Time in Months

Unit 6 Determining the Slope (Rate of Change) of a Line given the Coordinates of Two Points

[pic]

The symbol m is used in mathematics to represent the slope.

[pic]

To calculate the slope (rate of change) of a line, given the coordinates of two points, the following formula is applied:

[pic]

Example : Determine the slope (rate of change) of the line passing through the points

[pic] and [pic].

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Scale : x-axis: division = y-axis: division =

Example : Given the following two points in a line calculate, for each line, the slope (rate of change)

(a) P1(– 2 , 3) and P2(3 , 3). (b) P1(– 2 , 2) and P2(–2 , –3)

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Example : Given the following two points in a line calculate, for each line, the slope (rate of change)

(a) P1(– 3 , –1) and P2(2 , 2). (b) P1(– 1 , 3) and P2(3 , –2)

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Unit 7 Graphing a Line given its slope and the Coordinates of One Point

A displacement is positive when it is effected in the increasing direction of the axes and negative when it is measured in the decreasing direction of the axes.

To graph a line, given its slope and the coordinates of one of its points:

1. Plot the given point on a graduated Cartesian plane.

2. From this point, measure a vertical distance and a horizontal distance corresponding to the ratio given by the slope of this line:

• If the slope is positive, the two displacements will be measured in the same direction (positive or negative).

• If the slope is negative, one displacement will be measured in one direction of the axes of the Cartesian plane and the other in the opposite direction.

3. Identify the coordinates of the point determined in step 2.

4. Draw the line through these two points.

5. Verify the slope (rate of change), using these two points.

Unit 8 Determining the Slope (Rate of Change) of a Line given its Equation

To convert a fraction [pic] to a decimal , divide the numerator a by the denominator b. Thus [pic] becomes

3 ÷ 5 = 0.6

In any linear equation of the form y = m x + b, m represents the slope of the line and b, the y − intercept.

To determine the slope of a non – vertical line, given its equation:

1. Isolate the variable y to convert the equation to the form y = m x + b.

2. Establish the value of the coefficient m of the variable x as the value of the slope.

In any linear equation of the form Ax + By + C = 0, its slope is equal to [pic] if B ≠ 0.

To determine the slope of a line, given its equation:

1. Convert the equation to the form Ax + By + C = 0.

2. Calculate the value of [pic] if B ≠ 0.

In an equation of the form A x + B y + C = 0 , A represents the numerical coefficient of x , B the numerical coefficient of y , and C the constant. Thus in 3x + 2y – 8 =0 , A = 3 , B = 2 , and C = − 8.

To determine the slope of a line, given its equation:

1. Either write the equation in the form y = m x + b and establish the coefficient of m of the variable x as the slope.

2. or, write the equation in the form Ax + By + C = 0 and calculate m = [pic].

Unit 9 Determining the Equation of a Line given its Graph

Slope of the line [pic]

Equation of the line [pic] where m is the slope of the line, (x1 , y1) is a specific point on the line and (x , y) is any other point on the line.

In any proportion, the product of the extremes = the product of the means.

To determine the equation of a line :

1. Determine the slope by applying the formula [pic] or by using graphic methods.

2. Determine its equation by applying one of the following formulas :

a). y = m x + b if you know the y-intercept ; you then need only replace m and b by their respective values

b). [pic] if you know another point on the line. You then need only replace

x1 , y1 and m by their respective values, apply the fundamental property of proportions and express the equation in the form A x + B y + C = 0.

3. Verify the equation obtained by substituting the coordinates of a point in the line for the x and y variables.

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Scale : x-axis: division = y-axis: division =

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Scale : x-axis: division = y-axis: division =

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