Equations Of Lines Quiz! Math Trivia

Explanation
The equation of a line is typically written in the form y = mx + b, where m represents the slope and b represents the y-intercept. In this case, the slope is given as 8 and the y-intercept is (0, -2). Therefore, the equation of the line can be written as y = 8x - 2. This equation represents a line with a slope of 8 and a y-intercept of -2.

Explanation
The slope of a line represents the rate at which the line is ascending or descending. In the equation y = 2/3x + 5, the coefficient of x is 2/3, which indicates that for every 1 unit increase in x, the corresponding increase in y is 2/3 units. Therefore, the slope of the given line is 2/3.

Explanation
The slope of a line is determined by the coefficient of the x-term in the equation of the line. In this case, the equation is y = -4x + 9. The coefficient of the x-term is -4, which means that the slope of the line is -4.

Explanation
The y-intercept of a line is the point where the line crosses the y-axis. In the equation y = 1/5x + 2, the y-intercept can be found by setting x = 0. When x = 0, the equation becomes y = 2. Therefore, the y-intercept is the point (0, 2).

Explanation
The y-intercept of a line is the point where the line intersects the y-axis. In the equation y = -5x + 6, the y-intercept is the constant term, which is 6. Therefore, the y-intercept of the line is (0, 6).

Explanation
The equation y = 3/4x + 1 is parallel to y = 3/4x + 5 because they have the same slope of 3/4. The only difference is that the y-intercept is different, with y = 3/4x + 1 having a y-intercept of 1 instead of 5.

Explanation
The given equation is in slope-intercept form, y = mx + b, where m is the slope of the line. The slope of the given line is 3/4. A line perpendicular to this line will have a negative reciprocal slope. The negative reciprocal of 3/4 is -4/3. Therefore, the equation y = -4/3x + 1 represents a line that is perpendicular to the given line y = 3/4x + 5.

Explanation
The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) can be found using the slope-intercept form y = mx + b, where m is the slope of the line and b is the y-intercept.

To find the slope, we use the formula m = (y₂ - y₁) / (x₂ - x₁). Plugging in the given points (2, 3) and (4, 5), we get m = (5 - 3) / (4 - 2) = 2 / 2 = 1.

Now that we have the slope, we can substitute it into the slope-intercept form and choose any of the given points to find the y-intercept. Using (2, 3), we get 3 = 1(2) + b, which gives us b = 1.

Therefore, the equation of the line passing through (2, 3) and (4, 5) is y = x + 1.

Explanation
The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) can be found using the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. By calculating the slope between the given points, it can be determined that the slope is 2. Plugging in the coordinates of one of the points, (3, 5), into the slope-intercept form, the equation becomes y = 2x - 1. Therefore, the correct answer is y = 2x - 1.

Explanation
The equation of a line can be determined using the formula y = mx + b, where m represents the slope of the line and b represents the y-intercept. In this case, the line passes through the points (0, 4) and (3, 6). To find the slope, we can use the formula (y2 - y1) / (x2 - x1), which gives us (6 - 4) / (3 - 0) = 2/3. Plugging this slope into the equation y = mx + b and substituting one of the given points, we get 4 = (2/3)(0) + b. Solving for b, we find that b = 4. Therefore, the equation of the line is y = 2/3x + 4.

Explanation
The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) can be determined using the formula (y - y₁) = (y₂ - y₁) / (x₂ - x₁) * (x - x₁). In this case, the two points given are (4, 6) and (4, -2). Since the x-coordinate of both points is the same, x = 4, the equation of the line passing through these points is x = 4.

Explanation
The equation of a line passing through two points can be found using the slope-intercept form, y = mx + b. In this case, both points have the same y-coordinate, which means the line is horizontal. Therefore, the slope (m) is 0. Plugging the coordinates (3, 5) into the equation, we get 5 = 0(3) + b. Solving for b, we find that b = 5. So the equation of the line passing through (3, 5) and (2, 5) is y = 5.