Mathematics
Grade 8
15 min
Write a linear equation from a slope and y-intercept
Write a linear equation from a slope and y-intercept
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Introduction & Learning Objectives
Learning Objectives
Identify the slope and y-intercept from given information.
Recall the slope-intercept form of a linear equation (`y = mx + b`).
Substitute given slope and y-intercept values into the slope-intercept form.
Write a linear equation in slope-intercept form given its slope and y-intercept.
Explain the meaning of the slope and y-intercept in a real-world context.
Verify the correctness of a written linear equation by comparing it to the given information.
Ever wondered how apps calculate the cost of your ride or how much your savings grow each month? 💰 Linear equations are behind many of these predictions!
In this lesson, you'll learn a fundamental skill: how to write a linear equation when you know its slope (steepness) and where it crosses the y-axis (...
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Key Concepts & Vocabulary
TermDefinitionExample
Linear EquationAn equation whose graph is a straight line. It describes a relationship where one quantity changes at a constant rate with respect to another.The equation `y = 5x - 2` is a linear equation.
Slope (m)A measure of the steepness of a line, representing the rate of change. It tells you how much the y-value changes for every 1-unit change in the x-value.In the equation `y = 3x + 1`, the slope `m` is `3`. This means for every 1 unit `x` increases, `y` increases by 3 units.
Y-intercept (b)The point where a line crosses the y-axis. At this point, the x-coordinate is always 0. It often represents an initial value or starting point.In the equation `y = 3x + 1`, the y-intercept `b` is `1`. This means the line crosses the y-axis at the point `(0, 1)`.
Slope-Interc...
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Core Formulas
Slope-Intercept Form of a Linear Equation
`y = mx + b`
This is the standard form used to write a linear equation when you know the slope (`m`) and the y-intercept (`b`). The `x` and `y` represent any point `(x, y)` on the line.
Identifying Slope and Y-intercept for Substitution
The slope `m` is the rate of change, and the y-intercept `b` is the initial value or the y-coordinate when `x=0`.
When given a slope and a y-intercept, you directly substitute these values into the `m` and `b` positions in the `y = mx + b` formula. If the y-intercept is given as a point `(0, b)`, you use `b`.
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Challenging
A container is being filled with water at a constant rate. It starts with 5 liters of water. After 1 minute, it has 8 liters. What is the linear equation representing the volume of water 'y' after 'x' minutes?
A.y = 8x + 5
B.y = 5x + 8
C.y = 5x + 3
D.y = 3x + 5
Challenging
Line A has a slope of 4 and a y-intercept of -2. Line B has a rate of change of -2 and an initial value of 4. Which statement is true about the equations for Line A (`y_A`) and Line B (`y_B`)?
A.The equation for Line A is `y_A = -2x + 4` and for Line B is `y_B = 4x - 2`.
B.The equation for Line A is `y_A = 4x - 2` and for Line B is `y_B = -2x + 4`.
C.Both lines have the same equation.
D.The lines have the same slope but different y-intercepts.
Challenging
A student writes the equation `y = b + mx`. Is this equation equivalent to the standard slope-intercept form `y = mx + b`? Why or why not?
A.Yes, because the Commutative Property of Addition allows the terms to be in any order.
B.No, because the slope 'm' must always be written before the y-intercept 'b'.
C.No, because 'b' is now the slope and 'm' is the y-intercept.
D.Yes, but only if both 'm' and 'b' are positive numbers.
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