Introduction
Welcome to our blog article on Unit 4 Linear Equations Homework 1 Slope Answer Key. In this article, we will provide you with the answers to the homework questions related to slope in linear equations. Understanding slope is crucial in solving and graphing linear equations, and we are here to help you grasp the concept. Let's dive in!
Question 1: Finding the Slope
Question 1a
Given the equation y = 2x + 5, find the slope.
In this equation, the coefficient of x is 2, which represents the slope. Therefore, the slope is 2.
Question 1b
Given the equation y = -3x - 2, find the slope.
In this equation, the coefficient of x is -3, which represents the slope. Therefore, the slope is -3.
Question 1c
Given the equation y = 1/2x + 3, find the slope.
In this equation, the coefficient of x is 1/2, which represents the slope. Therefore, the slope is 1/2.
Question 2: Graphing Linear Equations
Question 2a
Graph the equation y = -2x + 4.
To graph this equation, we start by plotting the y-intercept, which is 4. Then, using the slope of -2, we can find another point on the line by moving 2 units down and 1 unit to the right from the y-intercept. Connect the two points to complete the line.
Question 2b
Graph the equation y = 3/4x - 1.
To graph this equation, we start by plotting the y-intercept, which is -1. Then, using the slope of 3/4, we can find another point on the line by moving 3 units up and 4 units to the right from the y-intercept. Connect the two points to complete the line.
Question 3: Finding the Equation from Two Points
Question 3a
Given the points (2, 5) and (-1, -3), find the equation of the line passing through these points.
To find the equation, we first need to find the slope using the formula: slope = (y2 - y1) / (x2 - x1). Substituting the values, we get slope = (-3 - 5) / (-1 - 2) = -8 / -3 = 8/3. Next, we can use the point-slope form of a line: y - y1 = m(x - x1). Substituting one of the points, let's say (2, 5), we get y - 5 = 8/3(x - 2). Simplifying the equation gives us y - 5 = 8/3x - 16/3. Finally, rearrange the equation to get the slope-intercept form: y = 8/3x - 16/3 + 15/3. Therefore, the equation is y = 8/3x - 1/3.
Question 3b
Given the points (-2, 4) and (3, 1), find the equation of the line passing through these points.
Following the same steps as in Question 3a, we find the slope to be (1 - 4) / (3 - (-2)) = -3/5. Using the point-slope form with the point (-2, 4), we get y - 4 = -3/5(x - (-2)). Simplifying the equation gives us y - 4 = -3/5x - 6/5. Rearranging the equation gives us y = -3/5x - 6/5 + 20/5. Therefore, the equation is y = -3/5x + 14/5.
Question 4: Parallel and Perpendicular Lines
Question 4a
Given the equation y = 2x + 3, find the equation of a line parallel to this line passing through the point (4, -1).
Since parallel lines have the same slope, the slope of the new line will also be 2. Using the point-slope form with the point (4, -1), we get y - (-1) = 2(x - 4). Simplifying the equation gives us y + 1 = 2x - 8. Rearranging the equation gives us y = 2x - 9. Therefore, the equation of the parallel line is y = 2x - 9.
Question 4b
Given the equation y = -3x + 2, find the equation of a line perpendicular to this line passing through the point (-2, 5).
Perpendicular lines have slopes that are negative reciprocals of each other. The slope of the new line will be 1/3. Using the point-slope form with the point (-2, 5), we get y - 5 = 1/3(x - (-2)). Simplifying the equation gives us y - 5 = 1/3x + 2/3. Rearranging the equation gives us y = 1/3x + 17/3. Therefore, the equation of the perpendicular line is y = 1/3x + 17/3.
Conclusion
Understanding slope and its applications in linear equations is essential in mathematics. In this article, we provided you with the answers to the homework questions related to slope in linear equations. We covered finding the slope, graphing linear equations, finding the equation from two points, and identifying parallel and perpendicular lines. We hope this article has helped you in your studies and improved your understanding of linear equations. Good luck with your future math endeavors!