Write The Slope-intercept Equation Of A Line Whose Graph Has A Slope Of M = − 1 M = -1 M = − 1 And Passes Through The Point ( 5 , 1 (5, 1 ( 5 , 1 ].Use Y = M X + B Y = Mx + B Y = M X + B To Find The Equation. Do Not Use Decimals In Your Answer. Express All Parts Of Your

by ADMIN 271 views

Introduction

The slope-intercept equation of a line is a mathematical representation of a line in the form of y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will discuss how to find the slope-intercept equation of a line given its slope and a point on the line.

Understanding the Slope-Intercept Equation

The slope-intercept equation of a line is given by the formula y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is a measure of how steep the line is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The y-intercept of a line is the point at which the line intersects the y-axis.

Finding the Slope-Intercept Equation

To find the slope-intercept equation of a line, we need to know the slope of the line and a point on the line. In this case, we are given that the slope of the line is m = -1 and the line passes through the point (5, 1).

Step 1: Write the Equation in Point-Slope Form

The point-slope form of a line is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. In this case, we can substitute the given point (5, 1) and the slope m = -1 into the equation to get:

y - 1 = -1(x - 5)

Step 2: Simplify the Equation

To simplify the equation, we can distribute the slope to the terms inside the parentheses:

y - 1 = -x + 5

Step 3: Add 1 to Both Sides

To isolate y, we can add 1 to both sides of the equation:

y = -x + 6

Conclusion

In this article, we discussed how to find the slope-intercept equation of a line given its slope and a point on the line. We used the point-slope form of a line and simplified the equation to find the slope-intercept equation of the line. The final equation is y = -x + 6.

Example Problems

Problem 1

Find the slope-intercept equation of a line whose graph has a slope of m = 2 and passes through the point (3, 4).

Solution

To find the slope-intercept equation of the line, we can use the point-slope form of a line:

y - 4 = 2(x - 3)

Simplifying the equation, we get:

y - 4 = 2x - 6

Adding 4 to both sides, we get:

y = 2x - 2

Problem 2

Find the slope-intercept equation of a line whose graph has a slope of m = -3 and passes through the point (2, 1).

Solution

To find the slope-intercept equation of the line, we can use the point-slope form of a line:

y - 1 = -3(x - 2)

Simplifying the equation, we get:

y - 1 = -3x + 6

Adding 1 to both sides, we get:

y = -3x + 7

Applications of the Slope-Intercept Equation

The slope-intercept equation of a line has many applications in mathematics and real-world problems. Some of the applications include:

  • Linear Programming: The slope-intercept equation is used to find the optimal solution to a linear programming problem.
  • Calculus: The slope-intercept equation is used to find the derivative of a function.
  • Physics: The slope-intercept equation is used to model the motion of an object.
  • Economics: The slope-intercept equation is used to model the demand and supply of a product.

Conclusion

Frequently Asked Questions

Q: What is the slope-intercept equation of a line?

A: The slope-intercept equation of a line is a mathematical representation of a line in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

Q: How do I find the slope-intercept equation of a line?

A: To find the slope-intercept equation of a line, you need to know the slope of the line and a point on the line. You can use the point-slope form of a line to find the equation.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.

Q: How do I simplify the point-slope form of a line?

A: To simplify the point-slope form of a line, you can distribute the slope to the terms inside the parentheses and then add or subtract the same value to both sides of the equation.

Q: What is the y-intercept of a line?

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

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

A: To find the y-intercept of a line, you can substitute x = 0 into the slope-intercept equation of the line.

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 calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I find the slope of a line?

A: To find the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: What are some applications of the slope-intercept equation?

A: The slope-intercept equation has many applications in mathematics and real-world problems, including linear programming, calculus, physics, and economics.

Q: Can I use the slope-intercept equation to find the equation of a circle?

A: No, the slope-intercept equation is only used to find the equation of a line. To find the equation of a circle, you need to use a different formula.

Q: Can I use the slope-intercept equation to find the equation of a parabola?

A: No, the slope-intercept equation is only used to find the equation of a line. To find the equation of a parabola, you need to use a different formula.

Q: What is the difference between the slope-intercept equation and the standard form of a line?

A: The slope-intercept equation is in the form y = mx + b, while the standard form of a line is in the form Ax + By = C.

Q: How do I convert the slope-intercept equation to the standard form of a line?

A: To convert the slope-intercept equation to the standard form of a line, you can multiply both sides of the equation by the denominator of the slope and then rearrange the terms.

Q: Can I use the slope-intercept equation to find the equation of a horizontal line?

A: Yes, the slope-intercept equation can be used to find the equation of a horizontal line. The equation of a horizontal line is in the form y = b, where b is the y-intercept.

Q: Can I use the slope-intercept equation to find the equation of a vertical line?

A: No, the slope-intercept equation cannot be used to find the equation of a vertical line. The equation of a vertical line is in the form x = a, where a is the x-intercept.

Conclusion

In this article, we answered some frequently asked questions about the slope-intercept equation of a line. We discussed how to find the slope-intercept equation of a line, how to simplify the point-slope form of a line, and how to find the y-intercept and slope of a line. We also discussed some applications of the slope-intercept equation and how to convert it to the standard form of a line.