Mathematics
Grade 8
15 min
Write equation for proportional relationships
Write equation for proportional relationships
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1
Introduction & Learning Objectives
Learning Objectives
Identify proportional relationships from tables, graphs, and word problems.
Calculate the constant of proportionality (k) from given data.
Write an equation in the form y = kx to represent a proportional relationship.
Interpret the constant of proportionality in the context of a real-world problem.
Use the equation y = kx to solve for unknown values in proportional relationships.
Distinguish between proportional and non-proportional linear relationships.
Ever wonder how much you'd earn if you worked for a certain number of hours at a fixed rate? 💰 Or how many ingredients you need if you want to double a recipe? 🍳
In this lesson, you'll learn how to write mathematical equations that describe these kinds of 'fair share' or 'scale...
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Key Concepts & Vocabulary
TermDefinitionExample
Proportional RelationshipA relationship between two quantities where their ratio is constant. This means that as one quantity increases or decreases, the other quantity changes by the same factor.If you earn $10 per hour, working 2 hours earns $20, and working 3 hours earns $30. The ratio of earnings to hours (10/1, 20/2, 30/3) is always 10.
Constant of Proportionality (k)The constant ratio between two quantities in a proportional relationship. It represents the unit rate or the value of y when x is 1.In the earnings example, the constant of proportionality (k) is $10 per hour. For every 1 hour, you earn $10.
Equation of a Proportional RelationshipA mathematical equation that describes a proportional relationship, always in the form y = kx, where 'y' and &#...
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Core Formulas
Formula for Constant of Proportionality
$k = \frac{y}{x}$
To find the constant of proportionality (k), divide the y-value by the corresponding x-value for any point (x, y) in the relationship (where x is not zero).
Equation for Proportional Relationship
$y = kx$
Once you have found the constant of proportionality (k), you can write the equation that represents the proportional relationship by substituting the value of k into this formula. This equation allows you to find any y-value given an x-value, or vice versa.
Identifying Proportionality from a Table
For all pairs $(x, y)$ in the table (where $x \neq 0$), the ratio $\frac{y}{x}$ must be constant.
To determine if a relationship shown in a table is proportional, calculate the ratio y/x for each pair of values. If...
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Challenging
A phone plan from Company A costs $45 for 15 GB of data. Company B's data cost is modeled by the equation y = 3.5x, where y is the cost and x is the gigabytes of data. Which equation represents Company A's cost, and which company offers a lower price per gigabyte?
A.y = 15x for A; Company B is cheaper.
B.y = 45x for A; Company B is cheaper.
C.y = 3.5x for A; They are the same price.
D.y = 3x for A; Company A is cheaper.
Challenging
A car's fuel efficiency is 30 miles per gallon. Write an equation to represent the relationship between the distance traveled in feet (y) and the number of gallons of gas used (x). (Note: 1 mile = 5,280 feet)
A.y = 30x
B.y = 176x
C.y = 5280x
D.y = 158400x
Challenging
A taxi service charges a $2.50 flat fee plus $1.50 per mile. A student graphs the total cost (y) versus the miles (x) and sees a straight line. Why can't this relationship be represented by an equation in the form y = kx?
A.Because the cost per mile is not constant.
B.Because the graph is not a straight line.
C.Because the relationship includes a flat fee, so the line does not pass through the origin (0,0).
D.Because the variables x and y are not related.
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