Solving Problems Involving the Equation

Oct 13, 2025 by recoveryshake.com 85 views

===========================================================

Introduction

In mathematics, a linear equation is a type of equation in which the highest power of the variable(s) is 1. The general form of a linear equation is $y = mx + b$ , where $m$ is the slope and $b$ is the $y$ -intercept. In this article, we will focus on finding the slope and $y$ -intercept of the equation $y = 3x$ .

Understanding the Slope

The slope of a linear equation is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In other words, it is the rate at which the line rises or falls as we move along the $x$ -axis.

Calculating the Slope

To calculate the slope of the equation $y = 3x$ , we can use the formula:

m = \frac{\text{rise}}{\text{run}}

In this case, the rise is 3 units (since the coefficient of $x$ is 3) and the run is 1 unit (since the variable is $x$ ). Therefore, the slope is:

m = \frac{3}{1} = 3

So, the slope of the equation $y = 3x$ is 3.

Understanding the $y$ -Intercept

The $y$ -intercept of a linear equation is the point at which the line intersects the $y$ -axis. It is the value of $y$ when $x$ is equal to 0.

Calculating the $y$ -Intercept

To calculate the $y$ -intercept of the equation $y = 3x$ , we can substitute $x = 0$ into the equation:

y = 3(0) = 0

So, the $y$ -intercept of the equation $y = 3x$ is 0.

Graphing the Equation

Now that we have found the slope and $y$ -intercept of the equation $y = 3x$ , we can graph the line. To do this, we can use the slope-intercept form of the equation, which is:

y = mx + b

In this case, the slope is 3 and the $y$ -intercept is 0. Therefore, the equation becomes:

y = 3x + 0

or simply:

y = 3x

We can graph this line by plotting the $y$ -intercept (0, 0) and then using the slope to find other points on the line.

Conclusion

Real-World Applications

The equation $y = 3x$ has many real-world applications. For example, it can be used to model the cost of goods sold, the revenue generated by a business, or the distance traveled by an object. In each of these cases, the slope of the equation represents the rate at which the cost, revenue, or distance changes.

Cost of Goods Sold

Suppose a business sells a product for $3 per unit. The cost of goods sold can be modeled using the equation $y = 3x$ , where $x$ is the number of units sold and $y$ is the total cost.

Revenue Generated

Suppose a business generates revenue of $3 per unit sold. The revenue generated can be modeled using the equation $y = 3x$ , where $x$ is the number of units sold and $y$ is the total revenue.

Distance Traveled

Suppose an object travels at a speed of 3 units per second. The distance traveled can be modeled using the equation $y = 3x$ , where $x$ is the time traveled in seconds and $y$ is the distance traveled.

Solving Problems Involving the Equation

The equation $y = 3x$ can be used to solve a variety of problems. For example, suppose we want to find the cost of 5 units sold. We can substitute $x = 5$ into the equation:

y = 3(5) = 15

So, the cost of 5 units sold is $15.

Conclusion

In this article, we have found the slope and $y$ -intercept of the equation $y = 3x$ . We have also graphed the line using the slope-intercept form of the equation. The slope of the equation is 3, and the $y$ -intercept is 0. This information can be used to graph the line and to solve problems involving the equation. The equation $y = 3x$ has many real-world applications, including modeling the cost of goods sold, the revenue generated by a business, and the distance traveled by an object.

===========================================================

Q: What is the slope of the equation $y = 3x$ ?

A: The slope of the equation $y = 3x$ is 3. This means that for every 1 unit increase in $x$ , the value of $y$ increases by 3 units.

Q: What is the $y$ -intercept of the equation $y = 3x$ ?

A: The $y$ -intercept of the equation $y = 3x$ is 0. This means that when $x$ is equal to 0, the value of $y$ is also equal to 0.

Q: How do I graph the equation $y = 3x$ ?

A: To graph the equation $y = 3x$ , you can use the slope-intercept form of the equation, which is $y = mx + b$ . In this case, the slope is 3 and the $y$ -intercept is 0. You can plot the $y$ -intercept (0, 0) and then use the slope to find other points on the line.

Q: What is the equation of the line that passes through the points (2, 6) and (4, 12)?

A: To find the equation of the line that passes through the points (2, 6) and (4, 12), we can use the slope-intercept form of the equation, which is $y = mx + b$ . We can calculate the slope using the formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points. Plugging in the values, we get:

m = \frac{12 - 6}{4 - 2} = \frac{6}{2} = 3

Now that we have the slope, we can use one of the points to find the $y$ -intercept. Let's use the point (2, 6):

6 = 3(2) + b

Solving for $b$ , we get:

b = 6 - 6 = 0

So, the equation of the line is $y = 3x$ .

Q: How do I find the equation of the line that passes through a given point and has a given slope?

A: To find the equation of the line that passes through a given point and has a given slope, you can use the point-slope form of the equation, which is $y - y_1 = m(x - x_1)$ . Plugging in the values, you can solve for $y$ to get the equation of the line.

Q: What is the equation of the line that passes through the point (0, 2) and has a slope of 2?

A: To find the equation of the line that passes through the point (0, 2) and has a slope of 2, we can use the point-slope form of the equation, which is $y - y_1 = m(x - x_1)$ . Plugging in the values, we get:

y - 2 = 2(x - 0)

Simplifying, we get:

y - 2 = 2x

Adding 2 to both sides, we get:

y = 2x + 2

So, the equation of the line is $y = 2x + 2$ .

Q: How do I find the equation of the line that passes through two given points?

A: To find the equation of the line that passes through two given points, you can use the slope-intercept form of the equation, which is $y = mx + b$ . You can calculate the slope using the formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points. Then, you can use one of the points to find the $y$ -intercept.

Q: What is the equation of the line that passes through the points (1, 3) and (3, 5)?

A: To find the equation of the line that passes through the points (1, 3) and (3, 5), we can use the slope-intercept form of the equation, which is $y = mx + b$ . We can calculate the slope using the formula:

m = \frac{5 - 3}{3 - 1} = \frac{2}{2} = 1

Now that we have the slope, we can use one of the points to find the $y$ -intercept. Let's use the point (1, 3):

3 = 1(1) + b

Solving for $b$ , we get:

b = 3 - 1 = 2

So, the equation of the line is $y = x + 2$ .

Conclusion

In this article, we have answered some frequently asked questions about the equation $y = 3x$ . We have discussed the slope and $y$ -intercept of the equation, as well as how to graph the line and find the equation of the line that passes through a given point and has a given slope. We have also provided examples of how to find the equation of the line that passes through two given points.