Y=Mx+B: Understanding B in Slope-Intercept Form. Looking for a clear explanation of what ‘b’ represents in the equation y=mx+b? WHAT.EDU.VN provides you with a simple, easy-to-understand breakdown, plus examples. Dive in to master linear equations, y-intercept, and slope!
1. Understanding the Meaning of y = mx + b
The equation y = mx + b is a fundamental concept in algebra known as the slope-intercept form of a linear equation. This form provides a straightforward way to represent and analyze straight lines on a coordinate plane. Let’s break down each component:
- y: Represents the vertical coordinate on the coordinate plane. It’s the dependent variable, meaning its value depends on the value of x.
- x: Represents the horizontal coordinate on the coordinate plane. It’s the independent variable.
- m: This is the slope of the line. The slope indicates the steepness and direction of the line. It represents the change in ‘y’ for every unit change in ‘x’. A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards.
- b: This is the y-intercept of the line. The y-intercept is the point where the line crosses the y-axis. It’s the value of ‘y’ when ‘x’ is zero.
Therefore, b in the equation y = mx + b specifically denotes the y-intercept of the line. It’s the point (0, b) on the coordinate plane. Think of it as the starting point of the line on the y-axis. To visualize this, imagine a line on a graph. The slope ‘m’ tells you how much the line rises or falls for each step you take to the right, and the y-intercept ‘b’ tells you where the line begins on the vertical axis. This form allows you to easily identify these two crucial characteristics of any straight line.
2. Delving Deeper: How to Find y = mx + b
The beauty of y = mx + b lies in its ability to be easily determined when you have certain information about a line. There are a few primary ways to find this equation:
2.1. When Given the Slope (m) and the Y-Intercept (b)
This is the most straightforward scenario. If you are given the slope ‘m’ and the y-intercept ‘b’, you can simply plug these values into the equation.
Example:
Suppose you know the slope of a line is 3 (m = 3) and the y-intercept is 2 (b = 2). Then, the equation of the line is simply:
y = 3x + 2
2.2. When Given the Slope (m) and a Point on the Line (x₁, y₁)
In this case, you have the slope but not the y-intercept. You can use the point-slope form of a linear equation to find the equation and then convert it to slope-intercept form.
The point-slope form is: y – y₁ = m(x – x₁)
- Substitute: Plug in the given slope ‘m’ and the coordinates of the point (x₁, y₁) into the point-slope form.
- Solve for y: Rearrange the equation to solve for ‘y’, which will put the equation in slope-intercept form (y = mx + b).
Example:
Find the equation of a line with a slope of -2 that passes through the point (1, 4).
- Substitute: y – 4 = -2(x – 1)
- Solve for y:
- y – 4 = -2x + 2
- y = -2x + 2 + 4
- y = -2x + 6
Therefore, the equation of the line is y = -2x + 6.
2.3. When Given Two Points on the Line (x₁, y₁) and (x₂, y₂)
If you are given two points, you need to first find the slope ‘m’ and then use either of the points with the point-slope form (as described above) to find the equation.
-
Calculate the Slope: Use the slope formula:
m = (y₂ – y₁) / (x₂ – x₁)
-
Use Point-Slope Form: Choose either point (x₁, y₁) or (x₂, y₂) and the calculated slope ‘m’, then plug these values into the point-slope form: y – y₁ = m(x – x₁)
-
Solve for y: Rearrange the equation to solve for ‘y’ and obtain the slope-intercept form.
Example:
Find the equation of the line that passes through the points (2, 3) and (4, 7).
-
Calculate the Slope:
m = (7 – 3) / (4 – 2) = 4 / 2 = 2
-
Use Point-Slope Form: Let’s use the point (2, 3):
y – 3 = 2(x – 2)
-
Solve for y:
- y – 3 = 2x – 4
- y = 2x – 4 + 3
- y = 2x – 1
Therefore, the equation of the line is y = 2x – 1.
3. Step-by-Step: Writing an Equation in Slope-Intercept Form
Here’s a detailed breakdown of how to convert different forms of linear equations into slope-intercept form (y = mx + b):
3.1. Starting from Standard Form (Ax + By = C)
The standard form of a linear equation is Ax + By = C, where A, B, and C are constants. To convert this to slope-intercept form:
-
Isolate the ‘y’ term: Subtract Ax from both sides of the equation:
By = -Ax + C
-
Solve for ‘y’: Divide both sides by B:
y = (-A/B)x + (C/B)
Now the equation is in the form y = mx + b, where:
- m = -A/B (the slope)
- b = C/B (the y-intercept)
Example:
Convert the equation 3x + 4y = 8 to slope-intercept form.
-
Isolate the ‘y’ term:
4y = -3x + 8
-
Solve for ‘y’:
y = (-3/4)x + 2
So, the slope is -3/4, and the y-intercept is 2.
3.2. Starting from Point-Slope Form (y – y₁ = m(x – x₁))
As mentioned earlier, the point-slope form is useful when you have a point (x₁, y₁) and the slope ‘m’. To convert this to slope-intercept form:
-
Distribute ‘m’: Expand the right side of the equation:
y – y₁ = mx – mx₁
-
Isolate ‘y’: Add y₁ to both sides:
y = mx – mx₁ + y₁
-
Simplify: Combine the constants -mx₁ and y₁ into a single constant, which represents the y-intercept ‘b’:
y = mx + (y₁ – mx₁)
So, the slope is ‘m’, and the y-intercept is (y₁ – mx₁).
Example:
Convert the equation y – 2 = 5(x – 1) to slope-intercept form.
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Distribute ‘5’:
y – 2 = 5x – 5
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Isolate ‘y’:
y = 5x – 5 + 2
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Simplify:
y = 5x – 3
So, the slope is 5, and the y-intercept is -3.
3.3. Starting from a Horizontal or Vertical Line
- Horizontal Line: A horizontal line has a slope of 0 and its equation is always in the form y = b, where ‘b’ is the y-intercept. There’s no ‘x’ term because the ‘y’ value is constant for all ‘x’ values.
- Vertical Line: A vertical line has an undefined slope, and its equation is always in the form x = a, where ‘a’ is the x-intercept. These equations cannot be expressed in the slope-intercept form (y = mx + b) because they do not have a ‘y’ term that can be isolated.
Example:
- The equation y = 4 represents a horizontal line with a y-intercept of 4.
- The equation x = -2 represents a vertical line with an x-intercept of -2.
4. y = mx + b: Solved Examples
Let’s solidify your understanding with a few more examples:
Example 1: Identifying Slope and Y-Intercept
Given the equation y = -4x + 7, identify the slope and y-intercept.
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Solution: By comparing the equation to y = mx + b, we can see that:
- Slope (m) = -4
- Y-intercept (b) = 7
This means the line has a negative slope, sloping downwards from left to right, and crosses the y-axis at the point (0, 7).
Example 2: Writing the Equation from a Graph
Imagine a line on a graph that passes through the point (0, -3) and rises 2 units for every 1 unit it runs to the right. Write the equation of this line.
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Solution:
- The y-intercept (b) is -3 because the line passes through (0, -3).
- The slope (m) is 2 because the line rises 2 units for every 1 unit of run.
- Therefore, the equation of the line is y = 2x – 3.
Example 3: Finding the Equation from Two Points
Find the equation of the line that passes through the points (-1, 2) and (3, -4).
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Solution:
-
Calculate the Slope:
m = (-4 – 2) / (3 – (-1)) = -6 / 4 = -3/2
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Use Point-Slope Form: Using the point (-1, 2):
y – 2 = (-3/2)(x – (-1))
-
Solve for y:
- y – 2 = (-3/2)(x + 1)
- y – 2 = (-3/2)x – 3/2
- y = (-3/2)x – 3/2 + 2
- y = (-3/2)x + 1/2
-
Therefore, the equation of the line is y = (-3/2)x + 1/2.
Example 4: Real-World Application
A taxi charges an initial fee of $3.00 plus $2.50 per mile. Write an equation to represent the total cost (y) of a taxi ride in terms of the number of miles (x).
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Solution:
- The initial fee of $3.00 is the y-intercept (b = 3).
- The cost per mile, $2.50, is the slope (m = 2.50).
- Therefore, the equation is y = 2.50x + 3.
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5. Practice Questions on y = mx + b
Test your understanding with these practice questions. Answers are provided below.
- What is the equation of a line with a slope of 5 and a y-intercept of -1?
- A line passes through the points (0, 4) and (2, 0). What is its equation in slope-intercept form?
- Convert the equation 2x – 3y = 6 to slope-intercept form.
- A rental car company charges $25 per day plus a one-time fee of $50. Write an equation representing the total cost (y) for renting a car for (x) days.
- What are the slope and y-intercept of the line represented by the equation y = -x + 8?
Answers:
- y = 5x – 1
- y = -2x + 4
- y = (2/3)x – 2
- y = 25x + 50
- Slope = -1, y-intercept = 8
6. FAQs on y = mx + b
Still have questions? Check out these frequently asked questions:
| Question | Answer |
|---|---|
| What does ‘b’ represent in the equation y = mx + b? | ‘b’ represents the y-intercept of the line. It’s the point where the line crosses the y-axis (the vertical axis) on a graph. |
| How can I find the value of ‘b’ if I know the slope and one point on the line? | You can use the point-slope form (y – y₁ = m(x – x₁)) and then convert it to slope-intercept form (y = mx + b) to find ‘b’. |
| Can ‘b’ be negative? | Yes, ‘b’ can be negative. A negative ‘b’ simply means the line crosses the y-axis at a point below the origin (0, 0). |
| What happens to the equation if ‘b’ is zero? | If ‘b’ is zero, the equation becomes y = mx, which means the line passes through the origin (0, 0). |
| Is y = mx + b used in real-life situations? | Absolutely! It’s used to model various linear relationships, such as the cost of a service, distance traveled at a constant speed, or the relationship between temperature scales. |
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