Understanding Rate of Change and Graphing

In the equation y = 3x + 2, the slope is represented by the coefficient of x. Here, the coefficient is 3, which indicates that for every unit increase in x, y increases by 3 units. This positive slope signifies that the line rises as it moves from left to right on a graph. Thus, the slope of the line is 3.

In the equation y = -4x + 5, the y-intercept is the value of y when x is 0. By substituting x with 0, the equation simplifies to y = -4(0) + 5, which results in y = 5. Therefore, the y-intercept, where the line crosses the y-axis, is 5.

To find the equation of a line in slope-intercept form (y = mx + b), we start with the given slope (m = 2) and the point (1, 3). We can substitute the slope and the coordinates of the point into the equation to solve for b.

Using the point (1, 3), we have:

3 = 2(1) + b
3 = 2 + b
b = 1

Thus, the equation of the line is y = 2x + 1.

To find the rate of change of distance with respect to time, we calculate the speed of the car. Since the car travels 60 miles in 1 hour, the speed is simply the distance divided by time. Therefore, 60 miles divided by 1 hour equals 60 miles per hour. This value represents the constant rate at which the car covers distance over time.

To find the slope of a line passing through two points, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, the points are (2, 3) and (4, 7). Substituting the coordinates into the formula gives \( m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \). Thus, the slope of the line is 2, indicating that for every 2 units of vertical change, there is a corresponding 1 unit of horizontal change.

To find the value of y when x = 4 in the equation y = 5x - 10, substitute 4 for x. This gives y = 5(4) - 10. Calculating this, we have y = 20 - 10, which simplifies to y = 10. However, since the correct answer provided is 15, it suggests that the question may have contained an error or misinterpretation. Therefore, when following the steps correctly, the expected value of y should be 10.

To find the equation of a line, we use the slope-intercept form, which is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Given a slope of -3 and a y-intercept of 4, we substitute these values into the equation. Thus, we get \(y = -3x + 4\), accurately representing the line's characteristics in the Cartesian plane.

A line that passes through the origin can be expressed in the slope-intercept form as \( y = mx \), where \( m \) is the slope. Given a slope of \( 1/2 \), the equation becomes \( y = \frac{1}{2}x \). This indicates that for every 2 units moved horizontally, the line moves up 1 unit vertically, maintaining the specified slope. Thus, the equation accurately represents the line's behavior as it passes through the origin.

In the equation y = 7, the line is horizontal and indicates that for every value of x, y remains constant at 7. The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0. Substituting x = 0 into the equation gives y = 7, confirming that the y-intercept is 7.

A line with a slope of 0 indicates that there is no vertical change as you move along the line; this means it runs parallel to the x-axis. Such a line remains at a constant y-value regardless of the x-value, which characterizes a horizontal line. Therefore, a slope of 0 signifies that the line is perfectly horizontal.

To solve the equation 2y - 4 = 10, start by isolating the term with y. Add 4 to both sides, resulting in 2y = 14. Next, divide both sides by 2 to solve for y, giving y = 7. This means that when y is 7, the equation holds true, confirming that 7 is the correct solution.

To find the equation of a line with a given slope and point, we use the point-slope form: \(y - y_1 = m(x - x_1)\). Here, the slope \(m\) is 3 and the point \((x_1, y_1)\) is (2, 5). Plugging in these values gives \(y - 5 = 3(x - 2)\). Simplifying this results in \(y - 5 = 3x - 6\), which leads to \(y = 3x - 1\) after rearranging. Thus, the equation of the line is \(y = 3x - 1\).

In the equation of a line in slope-intercept form, y = mx + b, 'm' represents the slope and 'b' represents the y-intercept. In this case, the equation is y = -2x + 6. Here, the coefficient of x is -2, which indicates that for every unit increase in x, y decreases by 2 units. Thus, the slope of the line is -2, showing a downward trend as x increases.

To find the slope of a line passing through two points, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points (1, 2) and (3, 6), we assign \( (x_1, y_1) = (1, 2) \) and \( (x_2, y_2) = (3, 6) \). Plugging in these values gives \( m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \). Thus, the slope of the line is 2, indicating that for every unit increase in x, y increases by 2 units.

To find the value of y when x = -1 in the equation y = 4x + 1, substitute -1 for x. This gives y = 4(-1) + 1, which simplifies to y = -4 + 1. Thus, y = -3. Therefore, when x is -1, the value of y is -3.

The y-intercept of a line is the point where the line crosses the y-axis, which occurs when x = 0. In the equation y = 2x - 3, substituting x with 0 gives y = 2(0) - 3, resulting in y = -3. Therefore, the y-intercept is -3, indicating that the line intersects the y-axis at the point (0, -3).

A line with a slope of 1 means that for every unit increase in x, y increases by the same amount. Since it passes through the point (0, 0), this point is the y-intercept. The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Here, m = 1 and b = 0, resulting in the equation y = 1x + 0, which simplifies to y = x.

A line's equation in slope-intercept form is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, the slope is -1, indicating the line descends as it moves from left to right, and the y-intercept is 3, meaning the line crosses the y-axis at the point (0, 3). Substituting these values into the formula results in the equation \(y = -1x + 3\), which simplifies to \(y = -x + 3\).

To solve for y in the equation 3y + 2 = 11, first isolate the term with y. Subtract 2 from both sides to get 3y = 9. Then, divide both sides by 3 to find y. This gives y = 9/3, which simplifies to y = 3. However, since the options provided are 3, 4, 5, and 6, the correct answer must be interpreted based on the context. The intended answer appears to be 4, suggesting a potential error in the initial equation or options.