Match Each Equation To An Equivalent Equation Written In Slope-intercept Form.1. \[$x - 3y = -6\$\] - \[$y = \frac{1}{3}x + 2\$\]2. \[$-x + \frac{1}{2}y = 3\$\] - \[$y = 2x + 6\$\]3. \[$2y - 6 = X\$\] -
Introduction
In mathematics, the slope-intercept form is a fundamental concept in algebra, particularly in linear equations. It is a way of expressing a linear equation in the form of y = mx + b, where m represents the slope and b represents the y-intercept. In this article, we will delve into the world of slope-intercept form and explore how to match each equation to its equivalent form.
What is Slope-Intercept Form?
The slope-intercept form is a way of expressing a linear equation in the form of y = mx + b, where:
- m is the slope of the line
- b is the y-intercept of the line
The slope-intercept form is useful because it allows us to easily identify the slope and y-intercept of a line, making it easier to graph and analyze the line.
How to Match Equations to Slope-Intercept Form
To match an equation to its equivalent slope-intercept form, we need to follow these steps:
- Isolate the variable y: The first step is to isolate the variable y on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by a constant.
- Simplify the equation: Once the variable y is isolated, simplify the equation by combining like terms and eliminating any fractions.
- Write the equation in slope-intercept form: Finally, write the equation in the form of y = mx + b, where m is the slope and b is the y-intercept.
Example 1: Matching the Equation to Slope-Intercept Form
Let's take the first equation: x - 3y = -6.
To match this equation to its equivalent slope-intercept form, we need to isolate the variable y.
x - 3y = -6
Subtract x from both sides:
-3y = -x - 6
Divide both sides by -3:
y = (1/3)x + 2
Therefore, the equivalent slope-intercept form of the equation is y = (1/3)x + 2.
Example 2: Matching the Equation to Slope-Intercept Form
Let's take the second equation: -x + (1/2)y = 3.
To match this equation to its equivalent slope-intercept form, we need to isolate the variable y.
-x + (1/2)y = 3
Add x to both sides:
(1/2)y = x + 3
Multiply both sides by 2:
y = 2x + 6
Therefore, the equivalent slope-intercept form of the equation is y = 2x + 6.
Example 3: Matching the Equation to Slope-Intercept Form
Let's take the third equation: 2y - 6 = x.
To match this equation to its equivalent slope-intercept form, we need to isolate the variable y.
2y - 6 = x
Add 6 to both sides:
2y = x + 6
Divide both sides by 2:
y = (1/2)x + 3
Therefore, the equivalent slope-intercept form of the equation is y = (1/2)x + 3.
Conclusion
In conclusion, matching equations to slope-intercept form is a crucial skill in algebra. By following the steps outlined in this article, you can easily match any equation to its equivalent slope-intercept form. Remember to isolate the variable y, simplify the equation, and write it in the form of y = mx + b. With practice, you will become proficient in matching equations to slope-intercept form and be able to analyze and graph lines with ease.
Frequently Asked Questions
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I match an equation to its equivalent slope-intercept form?
A: To match an equation to its equivalent slope-intercept form, you need to isolate the variable y, simplify the equation, and write it in the form of y = mx + b.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is represented by the value b in the slope-intercept form of a linear equation.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep the line is. It is represented by the value m in the slope-intercept form of a linear equation.
References
Glossary
- Slope: A measure of how steep a line is.
- Y-intercept: The point where a line intersects the y-axis.
- Slope-Intercept Form: A way of expressing a linear equation in the form of y = mx + b.
- Linear Equation: An equation that can be written in the form of y = mx + b.
Slope-Intercept Form: A Comprehensive Guide to Matching Equations ===========================================================
Q&A: Frequently Asked Questions about Slope-Intercept Form
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I match an equation to its equivalent slope-intercept form?
A: To match an equation to its equivalent slope-intercept form, you need to isolate the variable y, simplify the equation, and write it in the form of y = mx + b.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is represented by the value b in the slope-intercept form of a linear equation.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep the line is. It is represented by the value m in the slope-intercept form of a linear equation.
Q: How do I find the slope of a line?
A: To find the slope of a line, you need to use the slope-intercept form of a linear equation, which is y = mx + b. The slope is represented by the value m.
Q: What is the difference between the slope and the y-intercept?
A: The slope is a measure of how steep a line is, while the y-intercept is the point where the line intersects the y-axis.
Q: Can I have a negative slope?
A: Yes, you can have a negative slope. A negative slope means that the line is decreasing as x increases.
Q: Can I have a zero slope?
A: Yes, you can have a zero slope. A zero slope means that the line is horizontal and does not change as x increases.
Q: Can I have a fractional slope?
A: Yes, you can have a fractional slope. A fractional slope means that the line is not a whole number and can be expressed as a fraction.
Q: How do I graph a line in slope-intercept form?
A: To graph a line in slope-intercept form, you need to use the slope-intercept form of a linear equation, which is y = mx + b. You can use the slope and y-intercept to graph the line.
Q: Can I have a vertical line in slope-intercept form?
A: No, you cannot have a vertical line in slope-intercept form. A vertical line is a line that is parallel to the y-axis and has an undefined slope.
Q: Can I have a horizontal line in slope-intercept form?
A: Yes, you can have a horizontal line in slope-intercept form. A horizontal line is a line that is parallel to the x-axis and has a slope of zero.
Q: How do I find the equation of a line in slope-intercept form?
A: To find the equation of a line in slope-intercept form, you need to use the slope-intercept form of a linear equation, which is y = mx + b. You can use the slope and y-intercept to find the equation of the line.
Q: Can I have a line with a negative y-intercept?
A: Yes, you can have a line with a negative y-intercept. A negative y-intercept means that the line intersects the y-axis below the x-axis.
Q: Can I have a line with a fractional y-intercept?
A: Yes, you can have a line with a fractional y-intercept. A fractional y-intercept means that the line intersects the y-axis at a point that is not a whole number.
Q: How do I find the equation of a line with a negative slope and a negative y-intercept?
A: To find the equation of a line with a negative slope and a negative y-intercept, you need to use the slope-intercept form of a linear equation, which is y = mx + b. You can use the slope and y-intercept to find the equation of the line.
Q: Can I have a line with a negative slope and a fractional y-intercept?
A: Yes, you can have a line with a negative slope and a fractional y-intercept. A negative slope and a fractional y-intercept means that the line intersects the y-axis at a point that is not a whole number and is below the x-axis.
Q: How do I graph a line with a negative slope and a negative y-intercept?
A: To graph a line with a negative slope and a negative y-intercept, you need to use the slope-intercept form of a linear equation, which is y = mx + b. You can use the slope and y-intercept to graph the line.
Q: Can I have a line with a negative slope and a zero y-intercept?
A: Yes, you can have a line with a negative slope and a zero y-intercept. A negative slope and a zero y-intercept means that the line intersects the y-axis at the origin and is below the x-axis.
Q: How do I find the equation of a line with a negative slope and a zero y-intercept?
A: To find the equation of a line with a negative slope and a zero y-intercept, you need to use the slope-intercept form of a linear equation, which is y = mx + b. You can use the slope and y-intercept to find the equation of the line.
Q: Can I have a line with a negative slope and a fractional y-intercept and a negative slope?
A: Yes, you can have a line with a negative slope and a fractional y-intercept and a negative slope. A negative slope and a fractional y-intercept and a negative slope means that the line intersects the y-axis at a point that is not a whole number and is below the x-axis.
Q: How do I graph a line with a negative slope and a fractional y-intercept and a negative slope?
A: To graph a line with a negative slope and a fractional y-intercept and a negative slope, you need to use the slope-intercept form of a linear equation, which is y = mx + b. You can use the slope and y-intercept to graph the line.
Conclusion
In conclusion, the slope-intercept form is a fundamental concept in algebra, and it is used to express linear equations in the form of y = mx + b. The slope-intercept form is useful because it allows us to easily identify the slope and y-intercept of a line, making it easier to graph and analyze the line. We have answered many frequently asked questions about slope-intercept form, and we hope that this article has been helpful in understanding the concept of slope-intercept form.
References
Glossary
- Slope: A measure of how steep a line is.
- Y-intercept: The point where a line intersects the y-axis.
- Slope-Intercept Form: A way of expressing a linear equation in the form of y = mx + b.
- Linear Equation: An equation that can be written in the form of y = mx + b.