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| 12 Theory slides |
| 12 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are some 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.
The given linear equation shows the relationship between the variables x and y. Determine if the equation is written in standard form.
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. |
Tearrik is excited to buy school supplies. He plans to buy some cool pencils and notebooks.
The pencils he wants cost $2 each, and the notebooks he likes cost $5 each. He has $20 to spend. The following linear equation models this situation.y=0
Zero Property of Multiplication
Identity Property of Addition
LHS/2=RHS/2
x=0
Zero Property of Multiplication
Identity Property of Addition
LHS/5=RHS/5
The number of stationery purchased cannot be negative. In other words, stores do not sale a negative amount of products. This means that only positive values of x and y is considered in this context.
First, the x-intercept means that if Tearrik does not buy any notebooks, he can buy 10 pencils with all of his money. Whereas, the y-intercept means that if Tearrik does not buy pencils, he can buy 4 notebooks using all of his money.
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
LHS−y=RHS−y
Subtract terms
LHS+8=RHS+8
Commutative Property of Addition
Add terms
Rearrange equation
x=0
Zero Property of Multiplication
Subtract terms
LHS⋅(-1)=RHS⋅(-1)
Tearrik realizes at the stationery shop that the price of five pencil cases equals $10 less than the price of two school bags. The graph below shows the relationship between the price of the school bag and the price of the pencil case.
The y-intercept and the slope of the line is used to write the equation of the line in slope-intercept form. Let's first look at the y-intercept. Remember that the y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis.
The y-intercept of the line is 5. Now find the ratio of the change in y-values to the change in x-values to find the slope.
LHS⋅2=RHS⋅2
a⋅cb=ca⋅b
Cancel out common factors
Simplify quotient
LHS−2y+10=RHS−2y+10
Distribute -1
Commutative Property of Addition
Subtract terms
Rearrange equation
The following equation displays the relationship between the variables x and y in slope-intercept form. Rewrite the equation in slope-intercept form in standard form.
The number of equations in standard form that meets these properties is exactly one for each line. However, there are infinitely many equivalent equations in standard form. In other words, linear equations that describe the same line are equivalent. These equivalent equations can be written by using the Multiplication Property of Equality.
We can recall the standard form of a linear equation. Ax+ By= C In this form, A, B, and C are real numbers and A and B cannot both equal 0. With this information, we can conclude that two of the given equations are written in standard form. 7x+ 5y&= 6 8x+ 9y&= 40
We want to rewrite the equation in standard form. Let's first recall the standard form of a line. Ax+ By= C In this form, A, B, and C are real numbers and A and B cannot both equal 0. Remember that the value of A should be positive. Let's rearrange the equation in standard form.
Standard form of the given equation is 2x-4y=20.
Solve the following equations for the desired variable.
We substitute 2 for y. Then we can solve the equation for x by isolating it.
When y is 2, x is equal to 4.
We will substitute 4 for x and solve the equation for y.
The value of y is -3.
Remember that the x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. The ordered pair of the point where the line crosses the x-axis is denoted as ( x, 0). This is why we substitute 0 for y and solve the equation for x to find the x-intercept.
The equation intercepts the x-axis at x=4. Similar to finding the x-intercept, the y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis. The ordered pair of the point where the line crosses the y-axis is denoted as ( 0, y). This is why we substitute 0 for x and solve the equation for y to find the y-intercept.
The equation intercepts the y-axis at y=-26.
Let's begin by finding the x- and the y-intercepts of the linear equation. 4x+5y=20 We will substitute 0 for y and solve the equation for x to find the x-intercept.
The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. The y-coordinate of the point is equal to 0. This means that we should substitute 0 for y to find the x-intercept of the given equation. Then, solve for x.
An x-intercept of 5 means that the graph passes through the x-axis at the point ( 5,0).
Similarly, the y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis. The x-coordinate of the point at the y-intercept is 0. Therefore, we will substitute 0 for x to find the y-intercept.
A y-intercept of 4 means that the graph passes through the y-axis at the point (0, 4).
We found the x- and y- intercepts of the equation.
x-intercept | (5,0) |
---|---|
y-intercept | (0,4) |
We can now graph the equation by plotting the intercepts and connecting them with a line.
This graph corresponds to option A.
We model the relationship between time and distance for Emily's adventure by using the standard form of a line. 5x+3y=80 We are given that Emily's speed while running is 5 kilometers per hour. We substitute 5 for y — the variable that represents Emily's speed while running.
Emily's speed while biking is 13 kilometers per hour.