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- A linear relationship between two quantities means they are related like this: When one quantity changes by a certain amount, the other quantity always changes by a set amount. In a linear relationship, one quantity has a constant rate of change with respect to the other. The relationship is called linear because its graph is a line.
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- Overview
- What are linear relationships?
- Linear relationships
- Modeling real world scenarios
- Translating word problems
- Try it!
- Linear equations in slope-intercept form
- Linear functions
- Your turn!
- Things to remember
A guide to understanding linear relationships on the digital SAT
1.Review the basics of linear relationships
2.Practice writing linear equations based on word problems
3.Identify the important features of linear functions
The skills covered here will be important for the following SAT lessons:
•Graphs of linear equations and functions
A linear relationship is any relationship between two variables that creates a line when graphed in the xy -plane. Linear relationships are very common in everyday life.
[Example: Maya and Geoff's heights]
[Example: Tai's runs]
In this lesson, we'll:
1.Review the basics of linear relationships
2.Practice writing linear equations based on word problems
Linear equations can be used to represent the relationship between two variables, most commonly x and y . To form the simplest linear relationship, we can make our two variables equal:
y=x
By plugging numbers into the equation, we can find some relative values of x and y .
If we plot those points in the xy -plane, we create a line.
Every possible linear relationship is just a modification of this simple equation. We might multiply one of the variables by a coefficient or add a constant to one side of the equation, but we'll still be creating a linear relationship.
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Modeling with linear equations: gym membership & lemonade
Let's look at some examples!
A car with a price of $17,000 is to be purchased with an initial payment of $5,000 and monthly payments of $240 . Which of the following equations can be used to find the number of monthly payments, m , required to complete the purchase, assuming there are no taxes or fees? [Show me the translation!] The width of a rectangular vegetable garden is w feet. The length of the garden is 8 feet longer than its width. Which of the following expresses the perimeter, in feet, of the vegetable garden in terms of w ? [Show me the translation!] The concession stand at a high school baseball game sold bags of peanuts for $2.50 each and hot dogs for $3.00 each. If the concession stand brought in $196 and sold 42 hot dogs, how many bags of peanuts did the concession stand sell? [Show me the translation!]
What will we be asked to do in linear equations word problems?
On the test, we may be asked to: •Write our own equation based on the word problem •Write our own equation and then solve it •Solve a given equation based on the word problem
Try: identify parts of a linear equation
A helicopter, initially hovering 35 feet above the ground, begins to ascend at a speed of 16 feet per second. Write an equation that can be used to find t , the number of seconds it takes for the helicopter to reach 179 feet above the ground.
The total height, which everything else must add up to, is
feet.
The starting height of the helicopter is
feet.
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Constructing linear equations from context
Slope-intercept form
The slope-intercept form of a linear function, y=mx+b , where m and b are constants, tells us both the slope and the y -intercept of the line: •The slope is equal to m . [What's slope?] •The y -intercept is equal to b . [What's the y-intercept?]
Standard form
The standard form of a linear function, Ay+Bx=C , where, A , B , and C are constants, will often be used in word problem scenarios that have two inputs, instead of an input and an output. To find the slope or y -intercept of a line in standard form, it's often most convenient to convert the equation to slope-intercept form by isolating y .
What will we be asked to do in linear function word problems?
On the test, we may be asked to: •Write our own linear function based on the word problem (We may need to calculate the slope or y -intercept in more challenging questions.) •Identify the meaning of a value in a given function that models a scenario
Practice: write a linear equation
Tamika purchases a new mattress for $600 , which she will pay for with an initial payment of $150 and monthly installments of $30 . Which of the following equations can be used to find the number of monthly installments, m , required to complete the purchase, assuming there are no taxes or fees?
Choose 1 answer:
Choose 1 answer:
•(Choice A)
600=30m−150
slope=change in ychange in x=y2−y1x2−x1
The slope-intercept form of a linear equation, y=mx+b , tells us both the slope and the y -intercept of the line:
•The slope is equal to m .
•The y -intercept is equal to b .
We can write the equation of a line as long as we know either of the following:
•The slope of the line and a point on the line
Feb 22, 2024 · A linear relationship (or linear association) is a statistical term used to describe a straight-line relationship between two variables. Linear relationships can be expressed either...
- 1 min
A linear relationship is a type of function where two quantities have a direct, linear connection and create a straight line when graphed. Learn to describe a linear relationship between two...
- 5 min
A linear model is an equation that describes a relationship between two quantities that show a constant rate of change. We represent linear relationships graphically with straight lines. A linear model is usually described by two parameters: the slope, often called the growth factor or rate of change, and the ...
Nov 21, 2023 · A linear relationship is a relationship or connection between two variables that will produce a straight line when graphed. There will be times when the data points are scattered and do not...
A linear relationship is the simplest association to analyse between two quantitative variables. A straight line relationship between [latex]y[/latex] and [latex]x[/latex] can be written in a number of ways, such as [latex]y = a + bx[/latex] or [latex]y = mx + c[/latex].
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