Challenging Maths Equation Quiz

The climber is scaling the cliff at a rate of 124 feet per hour. After 3 hours, the climber would have only climbed 3 * 124 = 372 feet. Since the cliff is 400 feet tall, the climber does not reach the top after 3 hours.

The slope-intercept form of a line is given by y = mx + b, where m represents the slope of the line and b represents the y-intercept. In this case, m is given as 3 and b is given as 2. Therefore, the equation of the line in slope-intercept form is y = 3x + 2.

The slope-intercept form of a line is y = mx + b, where m represents the slope and b represents the y-intercept. In this case, the given slope is -1/5 and the y-intercept is 2/3. Therefore, the correct equation in slope-intercept form is y = -1/5x + 2/3.

The equation for 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. In this case, the slope can be calculated as (y2 - y1) / (x2 - x1), which is (7 - 3) / (4 - 2) = 4/2 = 2. Plugging in the slope and one of the points (2, 3) into the equation, we get y = 2x - 1. Therefore, the given equation y = 2x - 1 is correct for the line passing through the points (2, 3) and (4, 7).

The equation y = -4x + 18 is the correct answer because it is in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. In this equation, the slope is -4, which matches the given slope of -4. The point (5,-2) is on the line, so when we substitute x = 5 and y = -2 into the equation, we get -2 = -4(5) + 18, which simplifies to -2 = -20 + 18 and further simplifies to -2 = -2, which is true. Therefore, the equation y = -4x + 18 is the correct equation for the line that passes through the point (5,-2) and has a slope of -4.

The equation for the line passing through the points (-4,4) and (2,2) is not 1/2x + 3. To find the equation of a line, we can use the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. First, let's find the slope using the formula (y2 - y1) / (x2 - x1). Plugging in the coordinates, we get (2 - 4) / (2 - (-4)) = -2 / 6 = -1/3. Now, we can choose one of the points and substitute it into the equation to find the y-intercept. Using (-4,4), we get 4 = (-1/3)(-4) + b, which simplifies to 4 = 4/3 + b. Solving for b, we get b = 4 - 4/3 = 8/3. Therefore, the equation of the line passing through the points (-4,4) and (2,2) is y = (-1/3)x + 8/3. Since this equation is not equal to 1/2x + 3, the answer is False.

To write the equation for a line, you need to know the slope and have at least one point on the line. The slope represents the steepness of the line, while the point helps determine the line's position on the coordinate plane. With the slope and a point, you can use the point-slope form of a linear equation to write the equation for the line. This form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.

The given equation y - 4 = 3(x + 2) represents the point-slope form of the equation of a line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. In this case, the point (-2, 4) lies on the line and the slope is 3. Therefore, the equation y - 4 = 3(x + 2) is the correct point-slope form of the equation of the line.

The given answer, 5x + 2y = -10, is the equation of the line passing through the points (-2,0) and (0,-5) in standard form. In standard form, the equation of a line is written as Ax + By = C, where A, B, and C are constants. By rearranging the equation, we can see that the coefficients of x and y are 5 and 2 respectively, and the constant term is -10. Therefore, the equation 5x + 2y = -10 satisfies the conditions and represents the line passing through the given points.

The slope of a line that is perpendicular to another line can be found by taking the negative reciprocal of the slope of the given line. In this case, the given line has a slope of -2/3. Taking the negative reciprocal of -2/3 gives us 3/2. Therefore, the slope of the line that is perpendicular to the given line is 3/2.

To find the equation of a line perpendicular to y = 2/3x - 1, we need to determine the negative reciprocal of the slope. The slope of the given line is 2/3, so the negative reciprocal is -3/2.

Using the point-slope form of a linear equation, we can plug in the slope (-3/2) and the coordinates of the given point (4,2) to find the equation of the line.

y - 2 = -3/2(x - 4)

Simplifying the equation gives us:

y - 2 = -3/2x + 6

Rearranging the equation to the slope-intercept form gives us:

y = -3/2x + 4

Therefore, the correct answer is y = -3/2x + 4.

The slope in the linear model for this situation is 124. This is because the climber is scaling the cliff at a constant rate of 124 feet per hour. The slope represents the rate of change, in this case, the increase in height per hour.

The equation of a line can be found using the formula 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 (y2 - y1) / (x2 - x1) using the coordinates of the two given points. In this case, the slope is (2 - 8) / (-4 - 4) = -6 / -8 = 3/4. The y-intercept can be found by substituting the coordinates of one of the points into the equation and solving for b. Using the point (4,8), we get 8 = (3/4)(4) + b. Solving for b, we find that b = 5. Therefore, the equation of the line is y = 3/4x + 5.

The point-slope form of the equation of a line is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope of the line. In this case, the point (1, -5) is given and the slope is 3. Plugging these values into the point-slope form equation, we get y + 5 = 3(x - 1).

The given equation of the line is y = -3x - 2. To find the slope intercept equation of a line parallel to this line, we know that the slope of the new line will be the same as the slope of the given line, which is -3. Since the line passes through the point (3,-4), we can substitute the values of x and y into the slope intercept equation y = mx + c and solve for c. Using the point (3,-4), we get -4 = -3(3) + c. Solving for c, we find that c = 5. Therefore, the slope intercept equation of the line passing through the point (3,-4) and parallel to y = -3x - 2 is y = -3x + 5.

The slopes of perpendicular lines are opposite reciprocals. This means that if the slope of one line is m, then the slope of the perpendicular line is -1/m. This relationship holds true for any two lines that are perpendicular to each other. The concept of opposite reciprocals is important in geometry and can be used to determine if two lines are perpendicular to each other.

The slope of the line passing through the points (-3,0) and (3,6) can be calculated using the formula (y2 - y1) / (x2 - x1). Substituting the values, we get (6 - 0) / (3 - (-3)) = 6 / 6 = 1. The slope of the line y = -x - 2 is -1. Since the product of the slopes of two perpendicular lines is -1, we can conclude that the line passing through the given points is perpendicular to the line y = -x - 2. Therefore, the answer is True.

The standard form of the equation of a line is Ax + By = C, where A, B, and C are constants. To find the equation of a line that passes through (-4,3) with a slope of -2, we can use the point-slope form of the equation, which is y - y1 = m(x - x1). Plugging in the values (-4,3) and -2 for x1, y1, and m respectively, we get y - 3 = -2(x + 4). Simplifying this equation gives us y - 3 = -2x - 8. Rearranging the terms, we get 2x + y = -5, which is the standard form of the equation of the line.

The given equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. To convert it into standard form, Ax + By = C, where A, B, and C are integers, we need to eliminate the fraction. By multiplying every term by 3, we get -2x + 3y = 12, which is the standard form with integer coefficients.

The equation 4x + 5y = 60 represents the different amounts of pasta salad (x) and potato salad (y) that can be bought with a budget of $60. The equation shows that the total cost of the pasta salad (4x) and the total cost of the potato salad (5y) together should be equal to $60. The second equation 60 = 4x + 5y is just a rearrangement of the first equation and represents the same relationship between the variables.