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| 11 Theory slides |
| 13 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
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.
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.
y=0
Zero Property of Multiplication
Identity Property of Addition
LHS/3=RHS/3
x=0
Zero Property of Multiplication
Identity Property of Addition
LHS/5=RHS/5
Now it is time to plot the intercepts in a coordinate plane.
Lastly, draw a line passing through these points.
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 | (AC,0) | (0,BC) |
A=0, B=0 | The line is horizontal, y=BC, so it does not cross the x-axis. | (0,BC) |
A=0, B=0 | (AC,0) | The line is vertical, x=AC, so it does not cross the y-axis. |
y-intercept: (0,6)
y=0
Zero Property of Multiplication
Identity Property of Addition
LHS/3=RHS/3
x=0
Zero Property of Multiplication
Identity Property of Addition
LHS/4=RHS/4
Since the number of kilograms of fruit purchased cannot be negative, only positive values of x and y make sense in this context.
Job | Amount Paid Per Hour ($) | Amount LaShay Makes ($) |
---|---|---|
I | 7 | 7x |
II | 10 | 10y |
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.
LHS⋅15=RHS⋅15
Distribute 15
Commutative Property of Multiplication
ca⋅b=ca⋅b
ba=b/5a/5
ba=b/3a/3
1a=a
Multiply
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.
Start by writing the equation of the line in point-slope form. Then, convert it into the standard form.
From the given graph, the x- and y-intercepts can be identified.
Substitute (60,0) & (0,9)
Subtract terms
ba=b/3a/3
Put minus sign in front of fraction
x=0, y=9
Zero Property of Multiplication
Multiply
Identity Property of Addition
Rearrange equation
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−y1=m(x−x1) | y=mx+b |
Dominika’s Equationy−43=-43(x−12)
|
Ali’s Equationy=-43x+(-433)
|
However, Dominika's calculations are not entirely correct. Until the last step everything is correct. However, she made a mistake when factoring out -43.
Rewrite 36 as 3⋅12
ca⋅b=ca⋅b
Factor out -43
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.
We will begin by reviewing the three forms in which a linear equation can be written.
Form | Equations | Features |
---|---|---|
Slope-Intercept Form | y=mx+n | m is the slope and b is the y-intercept. |
Point-Slope Form | (y-y_1)=m(x-x_1) | m is the slope and (x_1,y_1) is the coordinate of any point on the line. |
Standard Form | Ax+By=C | A, B, and C are real numbers and A and B are not both zero. |
We can now determine the form of the given linear equations. Let's analyze the first equation. y+7 = - (x-21) ⇕ y-( - 7)= -1(x- 21) We can identify the point ( 21, - 7) and the slope m= - 1. This means that this equation is in point-slope form. Next, we will consider the second equation. 7x-y= 21 ⇕ 7x+(- 1)y=21 This one is in standard form, where 7, - 1, and 21 are real numbers. Finally, we can identify the form of our last equation. y= 21x + 7 This is in slope-intercept form, where -2 is the slope and 7 is the y-intercept.
Think of the point where the graph of an equation crosses the x-axis. The y-value of that ( x, y) coordinate pair is 0, and the x-value is the x-intercept. To find the x-intercept of the equation, substitute 0 for y and solve for x.
An x-intercept of - 6 means that the graph passes through the x-axis at the point ( - 6,0).
Find the y-intercept following a similar fashion. Consider the point where the graph of the equation crosses the y-axis. The x-value of the ( x, y) coordinate pair at the y-intercept is 0. Therefore, substituting 0 for x will give the y-intercept.
The y-intercept of the equation is 24, which means that the graph passes through the y-axis at the point (0, 24).
The given equation models the earnings from the performance, where x is the number of students and y is the number of adults. 2.5x+5y=600 To graph this equation, we will first find its intercepts. This can be done by substituting 0 for one variable and solving the equation for the other variable. Let's first substitute 0 for y and solve the equation for x.
The point (240,0) is the x-intercept, which means that 240 students can attend the performance if no adults attend. Next, we will find the y-intercept. To do so, we substitute 0 for x and solve for y.
The y-intercept is (0,120), which means that 120 adults can attend the performance if no students attend. Finally, we can use the intercepts to draw the graph. rc Equation: & 2.5x+5y = 600 x-intercept: & (240,0) y-intercept: & (0,120) Let's plot these points on a coordinate plane and connect them with a line.
We can only consider the first quadrant because a negative number of attendees makes no sense in this context.
This corresponds to option C.
Equations in standard form are written in a specific format. Ax+ By= C Here, A, B, and C are integers and A and B cannot both be zero simultaneously. We will use the Properties of Equality to rewrite the given equation in this form.
Now we can see that our equation follows the standard form. Ax+ B y&= C 8x+ (- 20)y&= - 15
Ali says the equation y=9x+1 can be written in standard form as 9x−y=1.
Let's examine Ali's solution.
In the first step, Ali subtracts 9x from both sides of the equation. Here, we can say that the Subtraction Property of Equality is correctly applied. Then, he multiplies the left-hand side of the equation by - 1, but forgets to multiply the same number on the other side of the equation.
Therefore, we can say that the Multiplication Property of Equality is not applied correctly. The answer is C.
Consider the standard form of a linear equation.
Ax+By=C
In this equation A, B, and C are real numbers, and A and B cannot be zero simultaneously. In this form, the variable terms should be on the left-hand side and the constant term should be on the right-hand side. Let's write the given equation in standard form using the Properties of Equality.
The diagram shows the graph of a linear equation.
We will first write the equation of the line in slope-intercept form. y = mx + b In this form, m is the slope of the line and b is the y-intercept. We can directly identify the slope and the y-intercept on the given graph.
The y-intercept b is - 1 and the slope m is 32. Let's substitute these values into the slope-intercept form. y = mx + b ⇓ y = 3/2x + ( - 1) Now, we need to rewrite this equation in standard form. To do so, we remove the fraction and then move the variable terms to the left-hand side of the equation.
The equation is in standard form, which corresponds to the equation in IV.
Equations in standard form are written in a specific format. Ax+ By= C In this form, A, B, and C are integers and A and B cannot both be zero. We will use the Properties of Equality to rewrite the given equation in this form. To get rid of the fraction, we need to multiply both sides by its denominator.
Now we can see that our equation follows the standard form. Ax+ By&= C 7x+ 3y&= - 15 Therefore, the correct answer is D.