Graphing and Writing a Linear Equation in Standard Form Explained

Assumption	$x -$ intercept	$y -$ intercept
$A \neq = 0,$ $B \neq = 0$	$(\frac{C}{A}, 0)$	$(0, \frac{C}{B})$
$A = 0,$ $B \neq = 0$	The line is horizontal, $y = \frac{C}{B},$ so it does not cross the $x -$ axis.	$(0, \frac{C}{B})$
$A \neq = 0,$ $B = 0$	$(\frac{C}{A}, 0)$	The line is vertical, $x = \frac{C}{A},$ so it does not cross the $y -$ axis.

Assumption

x -

intercept

y -

intercept

A \neq = 0,

B \neq = 0

(\frac{C}{A}, 0)

(0, \frac{C}{B})

A = 0,

B \neq = 0

The line is horizontal,

y = \frac{C}{B},

so it does not cross the

x -

axis.

(0, \frac{C}{B})

A \neq = 0,

B = 0

(\frac{C}{A}, 0)

The line is vertical,

x = \frac{C}{A},

so it does not cross the

y -

axis.

Kriz says the equation of the line given below is $2 x + 3 y = 18 .$ LaShay says that the equation of the line is $y = - \frac{2}{3} x + 6 .$

Who is correct?

We want to decide who is correct about the equation of the given line. Notice that Kriz's equation is in standard form and LaShay's equation is in slope-intercept form.

	Equation	Form of the Equation
Kriz	2x+3y=18	Standard Form
LaShay	y=- 23x+6	Slope-Intercept Form

We will write the equation of the given line in slope-intercept form and compare it with LaShay's equation. Additionally, we will convert the equation of the line to standard form and compare it with Kriz's equation. To start, we can find the slope of the line by selecting two points on the line and calculating the rise and run.

The rise is - 2 and the run is 3. Remember, the slope is the ratio of the rise to the run.

The slope of the line is - 23. Look at the graph to see the y-intercept of the line. The line intercepts the y-axis at (0,6). This means that the y-intercept is 6. We can write the equation of the line by substituting these values in slope-intercept form. y= mx+ b ⇓ y= -2/3x+ 6 The equation of the line in slope intercept form is y=- 23x+6. This means that LaShay is correct! Now we will convert the equation in standard form to see if Kriz is also right. We will write terms with the variables on one side of the equation and the constant on the other side. We will prefer the smallest possible integers for the coefficients.

The equation of the line in standard form is 2x+3y=18. This means that Kriz is also correct! We can conclude that both Kriz and LaShay are correct!

Heichi goes grocery shopping with a total of

$ 12

in his pocket. He discovers that one kilogram of oranges costs

$ 3,

while one kilogram of apples costs

$ 2 .

He decides to buy both oranges and apples using his available money. The following equation models this situation.

3 x + 2 y = 12

In this equation,

x

is the number of kilograms of oranges and

y

represents the number of kilograms of apples. Graph the equation. What do the

x -

and the

y -

intercept represent in this context?

Choose the option that best describes the situation.

We want to graph and interpret the x- and the y-intercept of the following equation. 3x+2x=12 The equation is written in standard form. We substitute 0 for x to find the y-intercept.

The y-intercept is 6. This means that the line of this equation intercepts the y-axis at (0,6). Now we can find the x-intercept by substituting 0 for y in the equation.

The line of this equation intercepts the x-axis at (4,0). Let's plot these points on the coordinate plane and graph a line that passes through these two points. Both the number of kilograms of orange and apple cannot be negative. This means that the line will be drawn only in the first quadrant of the coordinate plane.

Now we can interpret the x- and the y-intercepts. Keep in mind that the y-coordinate of the x-intercept is always 0. In this context, it represents the number of kilograms of oranges that can be purchased with the total amount of money.

Similarly, x-coordinate of y-intercept is 0. This can be interpreted as the amount of apples we can buy with our total money.

The best option to describe this situation is C.

The graph below shows the time Ignacio spends to complete two projects for a class.

Here are some statements about the time spent on finishing these projects.

A . B . C . D . Ignacio will spend 180 minutes to complete Project 1 if he decides to do only Project 1 . Ignacio completes two of the projects in 90 minutes . Ignacio will spend 40 minutes to complete Project 2 if he completes Project 1 in 50 minutes . Ignacio found that Project 1 is harder than Project 2 .

Which of these statements could be true?

We want to decide which of the given statements could be true for the given graph. Notice that the points on the line of the equation satisfies the equation. The x-coordinate of the points represents the time spent on Project 1, and the y-coordinate of the points represents the time spent on Project 2.

Note that the sum of the coordinates of the points is the same, 90. This indicates that the required time to complete the projects is 90 minutes. We can write an equation using this information. x+ y=90 Let's look at the given statements to find the correct one.

A. Ignacio will spend 180 minutes to complete Project 1 if he decides to do only Project 1.

This statement means that Ignacio allocates all of his time to Project 1. This results in no time available for Project 2. We can substitute 0 for y in the equation to reflect this.

We found that x equals 90 minutes. This indicates that the time that we have to complete Project 1 is 90 minutes, not 180. This means that the first statement is false. We can continue with the second statement.

B. Ignacio completes two of the projects in 90 minutes.

We wrote the equation of the given graph. x+y=90 This equation shows that the total time spends to complete two of the project is 90 minutes, so this statement is correct. Let's look at the third option.

C. Ignacio will spend 40 minutes to complete Project 2 if he completes Project 1 in 50 minutes.

We will substitute 50 for the time spends to complete Project 1 into the equation to determine the allotted time for Project 2.

We found that y equals 40. We can conclude that Ignacio will complete Project 2 in 40 minutes if he spends 50 minutes to complete Project 1. We found that option C is correct. Take a look at option D.

D. Ignacio found that Project 1 is harder than Project 2.

We do not have enough information to determine if this statement is true, so this option is not the one we are looking for.

Jordan has the option of working as a server and cook for a restaurant. She is paid $$ 8$ per hour while she works as a server. She is paid $$ 10$ per hour as a cook. Jordan is keeping track of how she earned $$ 300$ over her first few weeks.

Write an equation in standard form to represent the relationship between Jordan's working hours as a server and as a cook. In this equation, assign

x

as the number of hours she works as a server and

y

as the number of hours she works as a cook.

We want to write an equation in standard form to represent Jordan's working hours as a cook and as a server. We have information about the hourly pay rates for both positions.

When we multiply the hourly pay by the number of working hours, we can determine the total amount of money earned from that job.

	Server	Cook
Payment ($/h)	$ 8	$ 10
Number of Hour She Worked	x	y
Money She Earns	8* x	10* y

We found the money she earned by working as a server and as a cook. Also, we want the total money she earns from working at the restaurant be $300. This means that when we add up the money that she earned as a server and as a cook, we should get a total of $300. We can write the equation in standard form. 8x +10y=300

Standard Form of a Line

Catch-Up and Review

Comparing the Slope-Intercept Form and Standard Form

Standard Form of a Line

Recognizing the Standard Form of a Line

Graphing a Linear Equation in Standard Form

Extra

Stationery Shopping

Answer

Hint

Solution

Writing a Linear Equation in Standard Form

Preparing for School

Hint

Solution

Comparing Prices

Hint

Solution

Finding the Values of $A,$ $B,$ and $C$

Converting the Slope-Intercept Form of a Line to Standard Form

Exploring the Standard Form of a Line

Standard Form of a Line

Recommended exercises

	12 Theory slides
	12 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions