Graphing Linear Equations Unit Test

The y-intercept of a line is the point where the line crosses the y-axis. In this case, the y-intercept is 1. This means that when x is equal to 0, the corresponding y value is 1.

The slope of a line is a measure of how steep the line is. A positive slope means that the line is increasing as it moves from left to right. In other words, as the x-values increase, the y-values also increase. This can be visualized as a line that is slanting upwards from left to right on a graph.

Line D has an undefined slope because it is a vertical line. A vertical line has no change in the x-coordinate, resulting in a denominator of zero when calculating the slope using the formula (change in y / change in x). Since division by zero is undefined, the slope of a vertical line is undefined.

The line x = 5 is vertical because it is a vertical line passing through the x-coordinate 5. Vertical lines have a constant x-coordinate and can intersect any y-coordinate, but they do not have a slope. In this case, the line x = 5 is a vertical line that intersects the y-axis at multiple points, making it a vertical line.

The slope of a line can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates are (-3,7) and (-9,4). Plugging these values into the formula, we get (4 - 7) / (-9 - (-3)) = -3 / -6 = 1/2. Therefore, the slope of the line is 1/2.

Line B has a slope of 0 because a line with a slope of 0 is a horizontal line. In other words, the line does not rise or fall as it extends horizontally. The slope of a line is determined by the change in the y-coordinates divided by the change in the x-coordinates between any two points on the line. Since a horizontal line has the same y-coordinate for all points, the change in y is always 0, resulting in a slope of 0.

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The line y = -7 is horizontal because it has a fixed y-coordinate of -7, regardless of the value of x. In other words, all the points on this line have the same y-coordinate, while their x-coordinates can vary. This indicates that the line is parallel to the x-axis and does not have any vertical movement. Therefore, it can be concluded that the line y = -7 is horizontal.

The given equations are in the form of linear equations. The equation x - 5y = -5 can be rewritten as y = (1/5)x + 1, while the equation 5x - 25y = 75 can be rewritten as y = (1/5)x - 3. Both equations have the same slope of 1/5, indicating that the graphs of the equations are parallel.

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The given equations are 3x - y = 5 and 2x + 6y = 24. To determine the relationship between their graphs, we can rearrange both equations to slope-intercept form (y = mx + b). The first equation becomes y = 3x - 5, and the second equation becomes y = -1/3x + 4. The slopes of the two lines are 3 and -1/3, which are negative reciprocals of each other. When two lines have slopes that are negative reciprocals, they are perpendicular to each other. Therefore, the graphs of these equations are perpendicular.

The equation y = -5 represents a line parallel to the x-axis because the y-coordinate remains constant (-5) for all values of x. In other words, no matter what the value of x is, the y-coordinate will always be -5. This indicates that the line is horizontal and does not have any vertical change. Thus, it is parallel to the x-axis.

The equation y = (1/3)x - 3 represents a line that intersects the y-axis at -3 and has a slope of 1/3. Since the slope is not zero or undefined, the line will intersect the x-axis at exactly one point. Therefore, this equation represents a line that intersects at exactly one point.

The equation of a line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. In this case, the slope is given as 2. The point (3, -1) lies on the line, so we can substitute these values into the equation. Plugging in x = 3 and y = -1, we get -1 = 2(3) + b. Solving for b, we find b = -7. Therefore, the equation of the line that passes through the point (3, -1) and has a slope of 2 is y = 2x + 5.

The equation 2y = 5x + 1.5 is not perpendicular to y = 0.4x - 7 because the slopes of the two lines are not negative reciprocals of each other. The given equation has a slope of 5/2, while the equation y = 0.4x - 7 has a slope of 0.4. For two lines to be perpendicular, their slopes must be negative reciprocals of each other, meaning that when multiplied together, they equal -1. Since 0.4 * (5/2) does not equal -1, the line 2y = 5x + 1.5 is not perpendicular to y = 0.4x - 7.

The equation of a line that is parallel to y = -x + 4 will have the same slope, which is -1. Using the point-slope form of a linear equation, we can plug in the coordinates of the given point (6,-3) and the slope (-1) to find the equation of the line. The equation will be y - (-3) = -1(x - 6), which simplifies to y + 3 = -x + 6.

The equation to relate your income y to the number of lawns you mow x is y = 25x - 75. This equation represents the total income (y) you earn by mowing x number of lawns. The $75 is subtracted from the total income as it represents the initial cost of the lawn mower. The $25 per lawn is then multiplied by the number of lawns mowed to calculate the income from mowing lawns.