Convert $y - 3 = \frac{1}{2}(x + 5$\] To Slope-intercept Form.
Introduction
In mathematics, the slope-intercept form of a linear equation is a powerful tool for representing lines on a coordinate plane. It is written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore how to convert a given equation to slope-intercept form, using the equation y - 3 = \frac{1}{2}(x + 5) as an example.
What is Slope-Intercept Form?
The slope-intercept form of a linear equation is a way of writing an equation that highlights the slope (m) and the y-intercept (b) of the line. The slope (m) represents the rate of change of the line, while the y-intercept (b) represents the point where the line intersects the y-axis. The slope-intercept form is useful for graphing lines, finding the equation of a line given two points, and solving systems of linear equations.
Step 1: Distribute the Coefficient
To convert the given equation to slope-intercept form, we need to start by distributing the coefficient \frac{1}{2} to the terms inside the parentheses. This will give us:
y - 3 = \frac{1}{2}x + \frac{5}{2}
Step 2: Add 3 to Both Sides
Next, we need to add 3 to both sides of the equation to isolate the term with the variable (y). This will give us:
y - 3 + 3 = \frac{1}{2}x + \frac{5}{2} + 3
Simplifying the left-hand side, we get:
y = \frac{1}{2}x + \frac{5}{2} + 3
Step 3: Simplify the Right-Hand Side
Now, we need to simplify the right-hand side of the equation by combining the fractions. To do this, we need to find a common denominator, which is 2. We can rewrite the fractions as:
y = \frac{1}{2}x + \frac{5}{2} + \frac{6}{2}
Combining the fractions, we get:
y = \frac{1}{2}x + \frac{11}{2}
Step 4: Write the Equation in Slope-Intercept Form
Finally, we can write the equation in slope-intercept form by rearranging the terms:
y = \frac{1}{2}x + \frac{11}{2}
This is the slope-intercept form of the equation, where the slope (m) is \frac{1}{2} and the y-intercept (b) is \frac{11}{2}.
Conclusion
Converting an equation to slope-intercept form is a straightforward process that involves distributing the coefficient, adding or subtracting terms to isolate the variable, and simplifying the right-hand side. By following these steps, we can easily convert the equation y - 3 = \frac{1}{2}(x + 5) to slope-intercept form, which is y = \frac{1}{2}x + \frac{11}{2}. This form is useful for graphing lines, finding the equation of a line given two points, and solving systems of linear equations.
Examples and Applications
Here are a few examples and applications of converting equations to slope-intercept form:
- Graphing Lines: To graph a line, we need to know the slope (m) and the y-intercept (b). By converting an equation to slope-intercept form, we can easily find the slope and y-intercept, which can be used to graph the line.
- Finding the Equation of a Line Given Two Points: To find the equation of a line given two points, we need to find the slope (m) and the y-intercept (b). By converting an equation to slope-intercept form, we can easily find the slope and y-intercept, which can be used to find the equation of the line.
- Solving Systems of Linear Equations: To solve a system of linear equations, we need to find the solution that satisfies both equations. By converting the equations to slope-intercept form, we can easily find the solution by equating the y-values.
Tips and Tricks
Here are a few tips and tricks for converting equations to slope-intercept form:
- Distribute the Coefficient: When distributing the coefficient, make sure to multiply each term inside the parentheses by the coefficient.
- Add or Subtract Terms: When adding or subtracting terms, make sure to combine like terms and simplify the right-hand side.
- Simplify the Right-Hand Side: When simplifying the right-hand side, make sure to find a common denominator and combine the fractions.
Introduction
In our previous article, we explored how to convert a given equation to slope-intercept form, using the equation y - 3 = \frac{1}{2}(x + 5) as an example. In this article, we will answer some frequently asked questions about converting equations to slope-intercept form.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is a way of writing an equation that highlights the slope (m) and the y-intercept (b) of the line. The slope (m) represents the rate of change of the line, while the y-intercept (b) represents the point where the line intersects the y-axis.
Q: How do I convert an equation to slope-intercept form?
A: To convert an equation to slope-intercept form, you need to follow these steps:
- Distribute the coefficient to the terms inside the parentheses.
- Add or subtract terms to isolate the variable (y).
- Simplify the right-hand side by combining like terms and finding a common denominator.
Q: What is the difference between the slope-intercept form and the standard form of a linear equation?
A: The slope-intercept form (y = mx + b) and the standard form (Ax + By = C) are two different ways of writing a linear equation. The slope-intercept form highlights the slope (m) and the y-intercept (b), while the standard form highlights the coefficients (A, B, and C).
Q: Can I convert an equation to slope-intercept form if it is not in the standard form?
A: Yes, you can convert an equation to slope-intercept form even if it is not in the standard form. However, you may need to rearrange the terms and simplify the equation to get it into slope-intercept form.
Q: How do I find the slope (m) and the y-intercept (b) of a line given in slope-intercept form?
A: To find the slope (m) and the y-intercept (b) of a line given in slope-intercept form, you can simply read the values from the equation. For example, in the equation y = 2x + 3, the slope (m) is 2 and the y-intercept (b) is 3.
Q: Can I use the slope-intercept form to graph a line?
A: Yes, you can use the slope-intercept form to graph a line. By plotting the y-intercept (b) and using the slope (m) to find other points on the line, you can graph the line.
Q: Can I use the slope-intercept form to find the equation of a line given two points?
A: Yes, you can use the slope-intercept form to find the equation of a line given two points. By finding the slope (m) and the y-intercept (b) using the two points, you can write the equation of the line in slope-intercept form.
Q: Can I use the slope-intercept form to solve systems of linear equations?
A: Yes, you can use the slope-intercept form to solve systems of linear equations. By equating the y-values of the two equations, you can solve for the variables and find the solution to the system.
Conclusion
Converting equations to slope-intercept form is a powerful tool for representing lines on a coordinate plane. By following the steps outlined in this article, you can easily convert equations to slope-intercept form and use them to graph lines, find the equation of a line given two points, and solve systems of linear equations.