Express The Linear Equation In Slope-intercept Form:${ Y = \frac{7}{4}x + 5 }$

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Introduction

In mathematics, linear equations are a fundamental concept that can be expressed in various forms. One of the most common forms is the slope-intercept form, which is represented as y = mx + b, where m is the slope and b is the y-intercept. In this article, we will focus on expressing the linear equation in slope-intercept form, specifically the equation y = (7/4)x + 5.

Understanding the Slope-Intercept Form

The slope-intercept form is a way of expressing a linear equation in a more intuitive and visual way. The slope (m) represents the rate of change of the equation, while the y-intercept (b) represents the point where the equation intersects the y-axis. In the given equation y = (7/4)x + 5, the slope is 7/4 and the y-intercept is 5.

Expressing the Linear Equation in Slope-Intercept Form

To express the linear equation in slope-intercept form, we need to rewrite the equation in the form y = mx + b. In this case, the equation is already in the correct form, with the slope (7/4) and y-intercept (5) clearly defined.

Breaking Down the Equation

Let's break down the equation y = (7/4)x + 5 to understand its components:

  • The slope (7/4) represents the rate of change of the equation. This means that for every unit increase in x, the value of y increases by 7/4 units.
  • The y-intercept (5) represents the point where the equation intersects the y-axis. This means that when x = 0, the value of y is 5.

Graphing the Equation

To visualize the equation, we can graph it on a coordinate plane. The graph will be a straight line with a slope of 7/4 and a y-intercept of 5.

Real-World Applications

The slope-intercept form has many real-world applications, including:

  • Physics: The slope-intercept form is used to describe the motion of objects under constant acceleration.
  • Economics: The slope-intercept form is used to describe the relationship between variables such as supply and demand.
  • Computer Science: The slope-intercept form is used to describe the relationship between variables such as input and output.

Conclusion

In conclusion, expressing the linear equation in slope-intercept form is a fundamental concept in mathematics. The slope-intercept form provides a more intuitive and visual way of expressing a linear equation, with the slope representing the rate of change and the y-intercept representing the point where the equation intersects the y-axis. By understanding the slope-intercept form, we can better analyze and solve linear equations in various real-world applications.

Additional Resources

For further learning, here are some additional resources:

  • Mathway: A online math problem solver that can help you solve linear equations in slope-intercept form.
  • Khan Academy: A online learning platform that provides video lessons and practice exercises on linear equations in slope-intercept form.
  • MIT OpenCourseWare: A online resource that provides free online courses and materials on linear equations in slope-intercept form.

Frequently Asked Questions

Here are some frequently asked questions about expressing linear equations in slope-intercept form:

  • Q: What is the slope-intercept form? A: The slope-intercept form is a way of expressing a linear equation in the form y = mx + b, where m is the slope and b is the y-intercept.
  • Q: How do I express a linear equation in slope-intercept form? A: To express a linear equation in slope-intercept form, you need to rewrite the equation in the form y = mx + b, where m is the slope and b is the y-intercept.
  • Q: What is the y-intercept? A: The y-intercept is the point where the equation intersects the y-axis. It represents the value of y when x = 0.
    Frequently Asked Questions: Expressing Linear Equations in Slope-Intercept Form ================================================================================

Q: What is the slope-intercept form?

A: The slope-intercept form is a way of expressing a linear equation in the form y = mx + b, where m is the slope and b is the y-intercept. This form provides a more intuitive and visual way of expressing a linear equation, with the slope representing the rate of change and the y-intercept representing the point where the equation intersects the y-axis.

Q: How do I express a linear equation in slope-intercept form?

A: To express a linear equation in slope-intercept form, you need to rewrite the equation in the form y = mx + b, where m is the slope and b is the y-intercept. This can be done by rearranging the terms of the equation to isolate y on one side of the equation.

Q: What is the y-intercept?

A: The y-intercept is the point where the equation intersects the y-axis. It represents the value of y when x = 0. In the slope-intercept form, the y-intercept is represented by the constant term b.

Q: How do I find the slope of a linear equation?

A: To find the slope of a linear equation, you need to compare the coefficients of the x and y terms. The slope is the ratio of the coefficient of the x term to the coefficient of the y term. For example, in the equation y = 2x + 3, the slope is 2.

Q: How do I find the y-intercept of a linear equation?

A: To find the y-intercept of a linear equation, you need to set x = 0 and solve for y. This will give you the value of y when x = 0, which is the y-intercept.

Q: Can I have a negative slope?

A: Yes, you can have a negative slope. A negative slope means that the line slopes downward from left to right. For example, in the equation y = -2x + 3, the slope is -2.

Q: Can I have a fractional slope?

A: Yes, you can have a fractional slope. A fractional slope means that the slope is a fraction, such as 1/2 or 3/4. For example, in the equation y = (1/2)x + 3, the slope is 1/2.

Q: How do I graph a linear equation in slope-intercept form?

A: To graph a linear equation in slope-intercept form, you need to plot the y-intercept and then use the slope to find other points on the line. The slope tells you how many units to move up or down for every unit you move to the right.

Q: Can I have a horizontal line?

A: Yes, you can have a horizontal line. A horizontal line has a slope of 0, which means that it does not slope up or down. For example, in the equation y = 3, the slope is 0.

Q: Can I have a vertical line?

A: Yes, you can have a vertical line. A vertical line has an undefined slope, which means that it does not slope up or down. For example, in the equation x = 2, the slope is undefined.

Q: How do I find the equation of a line given two points?

A: To find the equation of a line given two points, you need to use the slope formula to find the slope of the line, and then use the point-slope form to find the equation of the line.

Q: Can I have a linear equation with no slope?

A: No, you cannot have a linear equation with no slope. A linear equation must have a slope, which represents the rate of change of the equation.

Q: Can I have a linear equation with a slope of infinity?

A: No, you cannot have a linear equation with a slope of infinity. A linear equation must have a finite slope, which represents the rate of change of the equation.

Conclusion

In conclusion, expressing linear equations in slope-intercept form is a fundamental concept in mathematics. By understanding the slope-intercept form, you can better analyze and solve linear equations in various real-world applications. We hope that this FAQ article has provided you with a better understanding of the slope-intercept form and how to use it to solve linear equations.