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6. Writing and Graphing Equations in Standard Form
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Writing and Graphing Equations in Standard Form

This lesson delves into the intricacies of two key mathematical topics: rewriting equations into standard form and understanding what the standard form of a linear equation means. It serves as a step-by-step guide, enriched with practical examples, to help you navigate these complex subjects. Ideal for students aiming to excel in algebra or geometry, as well as teachers looking for instructional lessons, this content is a valuable educational asset. The use cases extend beyond academics to fields like engineering, computer science, and economics, where understanding linear equations is essential.
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Writing and Graphing Equations in Standard Form
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In this lesson, a new form of linear equations will be introduced. Given a linear function in this form, its key components such as x- and y-intercepts will be identified and used to draw its graph. Additionally, converting between the forms of linear equations will be discussed.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Discussion

Standard Form of a Line

In the standard form of a line all x- and y-terms are on one side of the linear equation or function and the constant is on the other side.


Ax+By=C

In this form, A, B, and C are real numbers. It is important to know that A and B cannot both be 0. Different combinations of A, B, and C can represent the same line on a graph. It is preferred to use the smallest possible whole numbers for A, B, and C and it is also better if A is a positive number.

Line 3x-y=-3
Pop Quiz

Identifying the Standard Form of a Linear Equation

Consider the given linear equation that shows the relationship between the variables x and y. Determine whether the equation is written in standard form or not.

Linear equation written in different forms
Discussion

Graphing a Linear Function in Standard Form

A linear function written in standard form has quickly identifiable x- and y-intercepts. Since two points determine a line, this provides enough information to graph the function. Consider the following linear equation written in standard form. 3x+5y=30 The graph of this function can be drawn in two steps.
1
Find the Intercepts
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Begin by substituting y= 0 to find the x-intercept of the equation.
3x+5y=30
3x+5( 0)=30
Solve for x
3x+0=30
3x = 30
x = 10
The x-intercept is (10,0). The y-intercept can be found in a similar way. Substitute x= 0 into the equation and solve for y.
3x+5y=30
3( 0)+5y=30
Solve for y
0+5y=30
5y = 30
y = 6
The y-intercept is (0,6). rc Equation: & 3x+5y = 30 x-intercept: & (10,0) y-intercept: & (0,6)
2
Plot the Intercepts
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Now it is time to plot the intercepts in a coordinate plane.

Only intercepts of 3x+5y=30 are depicted as points: y-intercept at (0, 6) and x-intercept at (10, 0).
3
Draw the Line Passing Through the Intercepts
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Lastly, draw a line passing through these points.

Graph of the function 3x+5y=30 with y-intercept at (0,6) and x-intercept at (10,0).

Extra

General Formulas for the Intercepts of an Equation in Standard Form

Note that general formulas for the intercepts can be derived for any linear function written in standard form Ax+ By= C.

Assumption x-intercept y-intercept
A≠ 0, B≠ 0 (C/A,0) (0,C/B)
A= 0, B≠ 0 The line is horizontal, y= C B, so it does not cross the x-axis. (0,C/B)
A≠ 0, B= 0 (C/A,0) The line is vertical, x= C A, so it does not cross the y-axis.
Example

Using Intercepts to Graph a Linear Equation

On his way home from school, Ignacio stops to buy fruit at a market in his neighborhood. Oranges cost $3 per kilogram and apples cost $4 per kilogram. He has $24 to spend. The following linear equation models this situation. 3x+4y = 24 Here, x represents the number of kilograms of oranges and y represents the number of kilograms of apples.

a Find and interpret the intercepts of the linear equation.
b Graph the equation.

Answer

a x-intercept: (8,0)

y-intercept: (0,6)

Interpretation: See solution.
b

Hint

a To find the intercepts, substitute 0 for one of the variables and solve the equation for the other variable.
b Plot the intercepts and connect them with a line segment. Recall that the number of kilograms cannot be negative.

Solution

a The given equation models the total cost of the fruit, where x is the number of kilograms of oranges purchased and y is the number of kilograms of apples purchased.
3x+4y=24 The x-intercept will be found first. To do so, substitute 0 for y and solve the equation for x.
3x+4y=24
3x+4( 0)=24
Solve for x
3x+0=24
3x=24
x=8
The point (8,0) is the x-intercept, which means that if Ignacio does not buy any apples, he can buy 8 kilograms of oranges. Next, the y-intercept will be found. To do so, substitute 0 for x and solve for y.
3x+4y=24
3( 0)+4y=24
Solve for y
0+4y=24
4y=24
y=6
The y-intercept is (0,6). This means that if Ignacio does not buy any oranges, he can buy 6 kilograms of apples.
b In order to graph the equation, the intercepts found in the previous part can be used.

rc Equation: & 3x+4y = 24 x-intercept: & (8,0) y-intercept: & (0,6) Plot them on a coordinate plane and connect them with a line.

Intercepts of the equation and its graph

Since the number of kilograms of fruit purchased cannot be negative, only positive values of x and y make sense in this context.

Graph of the equation in the context of the situation
Example

Using the Standard Form of a Linear Equation to Model a Real-Life Situation

LaShay has one part-time job that she works at after school and a second part-time job that she works at on weekends. One pays $7 per hour and the other pays $10 per hour. She wants to make $350 per week. Amount Paid Per Hour [-0.9em] Job I: $7 Job II: $10

a Write a linear equation in standard form that describes this situation and draw its graph.
b If LaShay is allowed to work only 21 hours per week at the $10 per hour job, how many hours does she have to work per week at the other job in order to make $350?

Answer

a Equation: 7x+10y=350
Graph:
Graph of 7x+10y=350
b 20 hours per week

Hint

a To find the intercepts, substitute 0 for one of the variables and solve the equation for the other variable.
b Substitute the given value into the equation from Part A.

Solution

a Let x be the number of hours worked at Job I and y be the number of hours worked at Job II. Then, the amount of money LaShay can make from each part-time job can be written in terms of x and y.
Job Amount Paid Per Hour ($) Amount LaShay Makes ($)
I 7 7 x
II 10 10 y

Since LaShay wants to make $350 per week, the sum of 7x and 10y should be equal to 350. 7 x+10 y =350 To graph this equation, its intercepts will be found. Substitute y=0 to find the x-intercept and x=0 to find the y-intercept.

7x+10y=350
Operation x-intercept y-intercept
Substitution 7x+10( 0)=350 7( 0)+10y=350
Calculation x=50 y=35
Point (50,0) (0,35)

Now, plot the intercepts on a coordinate plane and connect them with a line segment. Since the number of hours worked cannot be negative, only positive values of x and y make sense.

Graph of 7x+10y=350
b Recall that the variable y in the equation written in Part A represents the number of hours worked at Job II. Therefore, by substituting 21 for y into the equation, the number of hours that LaShay needs to work per week at Job I can be found.
7x+10y = 350
7x +10( 21) = 350
Solve for x
7x + 210 = 350
7x = 140
x = 20
LaShay needs to work 20 hours per week at Job I in order to achieve her goal.
Discussion

Writing a Linear Equation in Standard Form

Any linear equation can be rewritten in standard form. Consider the following linear equation that is written in slope-intercept form. y = 4/5x + 2/3 Using the Properties of Equality, the equation can be rewritten in standard form. Ax+By=C Here, A, B, and C are real numbers and A and B cannot both be equal to 0. It can be noted that representing A, B, and C with the smallest possible integers is preferred, as well as A being positive.
1
Remove All Fractions
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When a linear equation contains fractions, the first step is to remove all fractions. The equation is multiplied by the least common denominator of the fractions. In this case, it is the product of 5 and 3, which is 15. Using the Multiplication Property of Equality, the equation can be written as follows.
y= 4/5x + 2/3
15y=(4/5x+2/3)15
Simplify right-hand side
15y=4/5x* 15+2/3* 15
15y=4/5* 15* x+2/3* 15
15y=4* 15/5x+2* 15/3
15y=4* 3/1x+2* 15/3
15y=4* 3/1x+2* 5/1
15y=4* 3x+2* 5
15 y = 12x + 10
2
Rearrange the Terms of the Equation
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Now, move the terms containing variables to the left-hand side of the equation. In addition to this, move the constant to the right-hand side using the Subtraction Property of Equality.
15 y = 12x + 10
- 12x + 15y = 10
Since a positive coefficient for x is preferable, the equation can be multiplied by - 1.
- 12x + 15y = 10
12x-15y = - 10
This equation is now in the standard form. Note that the values of A, B, and C are in their smallest possible integer forms.
Example

Writing a Linear Equation in Standard Form From a Graph

Jordan wants to buy some songs and movies online to enjoy after school. She can buy songs for $0.75 each and movies for $5 each. The graph represents the relationship between the number of songs purchased x and the number of movies purchased y.

Relationship between x and y
Write an equation in standard form that describes the relationship between x and y. Give the answer such that A, B, and C are the smallest possible integers and A is positive.

Hint

Start by writing the equation of the line in point-slope form. Then, convert it into the standard form.

Solution

From the given graph, the x- and y-intercepts can be identified.

Relationship between x and y
The intercepts are (60,0) and (9,0). When two points on a line are known, the point-slope form can be used to write the equation of the line. Recall that an equation in point-slope form follows a specific format. y- y_1= m(x- x_1) In this form, m is the slope and ( x_1, y_1) is a point on the graph of the line. To find the slope of the line, the intercepts can be used. Substitute them into the Slope Formula.
m = y_2-y_1/x_2-x_1
m=9- 0/0- 60
Simplify
m=9/- 60
m=3/- 20
m=- 3/20
Using the y-intercept ( 0, 9) — or any other point on the line — and the slope - 320, the equation of the line can be written. y- 9= -3/20(x- 0)⇔ y-9=- 3/20x Finally, this equation needs to be converted into the standard form. To do so, all fractions will be removed and the variable terms will be on the left-hand side of the equation.
y-9=- 3/20x
20y-180 = - 3x
3x+20y -180= 0
3x+20y = 180
This equation is in standard form.

Alternative Solution

Recall the standard form of a linear equation. Ax+By=C In the context of the problem, x is the number of songs Jordan can purchase and y is the number of movies she can purchase. If the values of A and B are considered as the costs of a song and a movie, respectively, then the equation shows the amount of money Jordan spends. Ax+By=C ⇓ 0.75x+5y = C Now, the value of C can be calculated by using one of the intercepts.

Relationship between x and y
The graph intercepts the axes at (60,0) and (0,9). For simplicity, the y-intercept (0,9) will be used.
0.75x+5y = C
0.75( 0) + 5 ( 9) = C
Solve for C
0+5(9) =C
0+45=C
45 = C
C=45
Therefore, the relationship between x and y can be expressed by the following equation. 0.75x+5y = 45 The number on the right hand-side 45 can be interpreted as Jordan's budget for her multimedia purchases. Finally, since it is preferred to rewrite the coefficients as the smallest possible integers, multiply the equation by 4.
0.75x+5y=45
3x+20y=180
Example

Converting Between the Forms of a Linear Equation

Dominika and Ali are working on an extra credit assignment after school. They have been given a linear equation written in standard form to solve. 3x+4y=- 33 They converted the equation into alternative forms of a linear equation.

Operations applied by Dominika and Ali
a What linear equation forms did Dominika and Ali write?
b Are their calculations correct? If not, describe and correct any mistakes.

Answer

a Dominika: Point-Slope Form
Ali: Slope-Intercept Form
b Are Ali's calculations right? Yes.
Are Dominika's calculations right? No, see solution

Hint

a A linear equation in slope-intercept form has the form y=mx+b. A linear equation in point-slope form has the form y-y_1=m(x-x_1).
b Pay close attention when factoring out a negative number.

Solution

a Consider the last lines of the steps.

The equation found by Dominika is written in point-slope form and the other equation is written in slope-intercept form.

Point-Slope Form Slope-Intercept Form
y- y_1 = m(x- x_1) y = mx+ b
Dominika's Equation [-0.7em] y- 3/4= -3/4(x- 12) Ali's Equation [-0.7em] y = -3/4x+ ( -33/4)
b It can be seen that Ali wrote the given equation in slope-intercept form correctly. He started by subtracting 3x from both sides of the equation. Then, he applied the Division Property of Equality without making a mistake.

However, Dominika's calculations are not entirely correct. Until the last step everything is correct. However, she made a mistake when factoring out - 34.

The given equation can be written correctly in point-slope form as follows.
y-3/4=-3/4x-36/4
Factor out -3/4
y-3/4=-3/4x-3* 12/4
y-3/4=-3/4x-3/4* 12
y-3/4=-3/4(x+12)
Pop Quiz

Finding the Values of A, B, and C

Consider the standard form of a linear equation. Ax+By= C In general, A, B, and C are real numbers. However, it is preferred for A, B, and C to be the smallest possible integers and for A to be positive. With this information in mind, write the values of A, B, and C by finding the equation of the given line in standard form. Give the answers such that A, B, and C are the smallest possible integers and A is positive.

Closure

Exploring the Standard Form of a Linear Equation

Throughout the lesson, the standard form of a linear equation has been discussed. Ax+By=C In general, A, B, and C are real numbers. However, it was previously noted that there are preferred properties for these numbers.

For each line there is exactly one equation in standard form that meets these properties. However, infinitely many equivalent linear equations in standard form can be obtained by using the Multiplication Property of Equality. Linear equations are equivalent if they describe the same line.

Equivalent equations
Writing and Graphing Equations in Standard Form
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