Example Question - slope and y-intercept
Here are examples of questions we've helped users solve.
Graphing a Line Using Point-Slope Form
To graph the line given by the equation \( y - 2 = \frac{-2}{5}(x - 4) \), you can follow these steps: 1. **Identify the slope and y-intercept:** The equation is in point-slope form, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. In this case, the slope (\( m \)) is \( \frac{-2}{5} \) and the point on the line is (4, 2). 2. **Plot the point on the graph:** Begin by plotting the point (4, 2) on your graph paper. This will be a point through which the line passes. 3. **Use the slope to find another point:** The slope \( \frac{-2}{5} \) means you move 2 units down and 5 units to the right from the point (4, 2) to find another point on the line. You could also move 2 units up and 5 units to the left if you wish to go in the opposite direction. 4. **Draw the line:** Connect the two points with a straight line. This line will extend infinitely in both directions. It's important to be precise and to plot points correctly. If you have graph paper, use a ruler to ensure that the line passes through the points accurately and extends across the grid.
Solving Linear Equations to Isolate a Variable
This equation is a linear equation in two variables, x and y. To solve it, we want to isolate y on one side. Here's how you can do it step-by-step: Given: \[ y + 2 = \frac{7}{3}(x + 1) \] Step 1: Distribute the \(\frac{7}{3}\) across the parenthesis to both x and 1. \[ y + 2 = \frac{7}{3}x + \frac{7}{3} \] Step 2: We want to get y on its own, so we need to subtract 2 from both sides of the equation to move the constant term on the y side to the other side. \[ y = \frac{7}{3}x + \frac{7}{3} - 2 \] Step 3: Combine the constant terms on the right-hand side. Since \(\frac{7}{3}\) is the same as \(2\frac{1}{3}\), you can subtract 2 from it. \[ y = \frac{7}{3}x + \frac{1}{3} \] Now y is expressed in terms of x. The equation here represents a line with a slope of \(\frac{7}{3}\) and a y-intercept at \(\frac{1}{3}\).
Solving Linear Equations in Slope-Intercept Form
The equation in the image is a linear equation in slope-intercept form, which is generally expressed as y = mx + b, where m is the slope and b is the y-intercept. To solve the equation y - 1 = -3(x - 5), we will simplify and put it in slope-intercept form. Here are the steps: 1. Distribute -3 into the parentheses: y - 1 = -3 * x + 3 * 5 y - 1 = -3x + 15 2. Add 1 to both sides to isolate y: y = -3x + 15 + 1 y = -3x + 16 Now the equation is in slope-intercept form y = -3x + 16, where the slope (m) is -3 and the y-intercept (b) is 16.
Graphing a Line in Slope-Intercept Form
Based on the image provided, the equation of the line is \(y + 3 = \frac{9}{4}(x + 4)\). To graph this line, it's typically easiest to write it in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. Let's rewrite the given equation in slope-intercept form: \[ \begin{align*} y + 3 &= \frac{9}{4}(x + 4) \\ y &= \frac{9}{4}x + \frac{9}{4} \cdot 4 - 3 \\ y &= \frac{9}{4}x + 9 - 3 \\ y &= \frac{9}{4}x + 6 \end{align*} \] Now we have the equation in slope-intercept form, with a slope \(m=\frac{9}{4}\) and a y-intercept \(b=6\). To graph this line: 1. Start at the y-intercept (0,6) on the graph. 2. Use the slope to find another point. The slope is the rise over the run, meaning you go up 9 units and right 4 units from the y-intercept (or down 9 and left 4, for a point in the opposite direction). 3. Plot the second point. 4. Draw a line through the two points, extending it across the graph. This straight line represents the equation \(y = \frac{9}{4}x + 6\).
Writing Linear Equations from a Table
To write the linear equation based on the provided table, we need to determine the slope and the y-intercept of the line. The slope (m) is the change in y divided by the change in x. Let's calculate it using the values from the table: m = (change in y) / (change in x) Looking at the y-values, each time x increases by 1, y decreases by 5 (for example, from -74 to -79, and from -79 to -84, etc.). So, our slope is -5. Now we need to determine the y-intercept (b), which is the value of y when x is 0. To find the y-intercept, we can use one of the points and the slope to write the equation. Let's take the first point (3, -74). y = mx + b -74 = -5(3) + b To find b, we solve for it: -74 = -15 + b b = -74 + 15 b = -59 Now we have both our slope and our intercept. The linear equation in the form y = mx + b is: y = -5x - 59 This is the equation that represents the rule shown in the table.
Finding Equation of a Line and Inequalities
To find the equation of the line, you need to find its slope and y-intercept. From the graph, you can see the line crosses the y-axis at the point (0, 1), which means the y-intercept is 1. To find the slope, you can pick two points on the line. The line passes through the points (0, 1) and (-2, -1) which can be clearly seen on the graph. The slope (m) is calculated as the rise over the run: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 1}{-2 - 0} = \frac{-2}{-2} = 1. \] So, the slope is 1. With a slope of 1 and a y-intercept of 1, the equation of the line in slope-intercept form (y = mx + b) is: \[ y = 1x + 1 \] or simply: \[ y = x + 1. \] For inequalities: a) Since the line is solid, this means that the inequality includes equality; that is, points on the line satisfy the inequality. Since the area below the line is shaded, the inequality represents all points where y is less than or equal to x + 1. Therefore, the inequality for part a) is: \[ y \leq x + 1. \] b) and c) are not provided explicitly in the question, but typically they would be related to different shading regions if provided. If the shaded region was above the line for part b), for example, the inequality would be y ≥ x + 1. If the shaded region for part c) omitted points on the line, the inequality would use strict inequality signs (< or >) rather than ≤ or ≥.
Equation of a Line and Inequalities in Shaded Regions
I'll help you to determine the equation of the line shown in part a of the image, and then we can use this information to write inequalities for the shaded regions shown in parts b and c. To find the equation of a line, we need to identify two key components: the slope and the y-intercept. In the coordinate grid provided, the line crosses the y-axis at (0, -2), which means the y-intercept is -2. The line also looks as if it passes through another point with integer coordinates, which we can use to determine the slope. I see that the line goes through (2, 2) along with (0, -2). The slope (\(m\)) is the change in y over the change in x when moving from one point to another: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-2)}{2 - 0} \] \[ m = \frac{4}{2} = 2 \] So the slope of the line is 2. Now that we have both the slope and the y-intercept, we can write the equation of the line in slope-intercept form (y = mx + b): \[ y = 2x - 2 \] For part b, if the graph is shaded above this line, then the inequality representing the shaded region would use a "greater than" symbol, because the y-values are greater than the y-values on the line for any given x-value. The inequality for the graph in part b would be: \[ y > 2x - 2 \] For part c, if the graph is shaded below this line, then the inequality representing the shaded region would use a "less than" symbol, because the y-values are less than the y-values on the line for any given x-value. The inequality for the graph in part c would be: \[ y < 2x - 2 \]
Calculating Slope and Y-Intercept from Graph
To find the slope and y-intercept of the line in the graph, we can use the following steps: **Slope:** The slope of a line is calculated by the rise over run. This means you need to determine how many units the line goes up (or down) for each unit it goes to the right. Looking at the graph, if we take two points on the line, for example, the point where it crosses the y-axis (the y-intercept) and any other point on the line, we can calculate the slope. From the graph, it looks like the line crosses the y-intercept at (0, 3) and goes through another point at (2, 7). Slope \( m = \frac{{change \, in \, y}}{{change \, in \, x}} = \frac{{y_2 - y_1}}{{x_2 - x_1}} \) So here, with our points (0, 3) and (2, 7): \( m = \frac{{7 - 3}}{{2 - 0}} \) \( m = \frac{4}{2} \) \( m = 2 \) The slope of the line is 2. **y-intercept:** The y-intercept is the point where the line crosses the y-axis. By inspecting the graph, the line crosses the y-axis at the point (0, 3). So the y-intercept is 3. Putting it all together, the slope is 2, and the y-intercept is 3.
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