Master Rewriting Equations in Standard Form and Understand the Standard Form of a Linear Equation

y= 0

3x+5( 0)=30

▼

Solve for x

Zero Property of Multiplication

3x+0=30

Identity Property of Addition

3x = 30

.LHS /3.=.RHS /3.

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

x= 0

3( 0)+5y=30

▼

Solve for y

Zero Property of Multiplication

0+5y=30

Identity Property of Addition

5y = 30

.LHS /5.=.RHS /5.

y = 6

The y-intercept is (0,6). rc Equation: & 3x+5y = 30 x-intercept: & (10,0) y-intercept: & (0,6)

Plot the Intercepts

Now it is time to plot the intercepts in a coordinate plane.

Draw the Line Passing Through the Intercepts

Lastly, draw a line passing through these points.

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.

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

y= 0

3x+4( 0)=24

▼

Solve for x

Zero Property of Multiplication

3x+0=24

Identity Property of Addition

3x=24

.LHS /3.=.RHS /3.

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

x= 0

3( 0)+4y=24

▼

Solve for y

Zero Property of Multiplication

0+4y=24

Identity Property of Addition

4y=24

.LHS /4.=.RHS /4.

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.

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

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:

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.

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

y= 21

7x +10( 21) = 350

▼

Solve for x

Multiply

7x + 210 = 350

SubEqn

LHS-210=RHS-210

7x = 140

.LHS /7.=.RHS /7.

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.

Remove All Fractions

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

LHS * 15=RHS* 15

15y=(4/5x+2/3)15

▼

Simplify right-hand side

Distr

Distribute 15

15y=4/5x* 15+2/3* 15

CommutativePropMult

Commutative Property of Multiplication

15y=4/5* 15* x+2/3* 15

MoveRightFacToNum

a/c* b = a* b/c

15y=4* 15/5x+2* 15/3

ReduceFrac

a/b=.a /5./.b /5.

15y=4* 3/1x+2* 15/3

ReduceFrac

a/b=.a /3./.b /3.

15y=4* 3/1x+2* 5/1

DivByOne

a/1=a

15y=4* 3x+2* 5

Multiply

15 y = 12x + 10

Rearrange the Terms of the Equation

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

SubEqn

LHS-12x=RHS-12x

- 12x + 15y = 10

Since a positive coefficient for x is preferable, the equation can be multiplied by - 1.

- 12x + 15y = 10

LHS * (- 1)=RHS* (- 1)

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.

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.

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

SubstitutePoints

Substitute ( 60,0) & ( 0,9)

m=9- 0/0- 60

▼

Simplify

SubTerms

Subtract terms

m=9/- 60

ReduceFrac

a/b=.a /3./.b /3.

m=3/- 20

MoveNegDenomToFrac

Put minus sign in front of fraction

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

LHS * 20=RHS* 20

20y-180 = - 3x

AddEqn

LHS+3x=RHS+3x

3x+20y -180= 0

AddEqn

LHS+180=RHS+180

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.

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

SubstituteII

x= 0, y= 9

0.75( 0) + 5 ( 9) = C

▼

Solve for C

Zero Property of Multiplication

0+5(9) =C

Multiply

0+45=C

Identity Property of Addition

45 = C

RearrangeEqn

Rearrange equation

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