Section 5.4 - Using Points To Determine The Equation Of A Line

1) Determine the equation of a line that passes through M(0, -6), with a slope of 2/3

Y = 2/3x + 4

Y = 2/3x - 6

Y = 4x + 2/3

Y = -6x + 2/3

The equation of a line can be determined using the point-slope formula, which states that 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 (0, -6) is given and the slope is 2/3. Plugging in these values into the point-slope formula, we get y - (-6) = (2/3)(x - 0), which simplifies to y + 6 = (2/3)x. By rearranging the equation, we get y = (2/3)x - 6, which matches the given answer.

Explanation

The equation of a line can be determined using the point-slope formula, which states that 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 (0, -6) is given and the slope is 2/3. Plugging in these values into the point-slope formula, we get y - (-6) = (2/3)(x - 0), which simplifies to y + 6 = (2/3)x. By rearranging the equation, we get y = (2/3)x - 6, which matches the given answer.

2)

Determine which equation is the equation for the line shown in the graph

Y = x - 1

Y = 1/2x - 1

Y = -1/2x - 1

Y = -1/2x + 1

The equation for the line shown in the graph can be determined by comparing the slope and y-intercept values of the given equations with the slope and y-intercept values of the line in the graph. The line in the graph has a slope of -1/2 and a y-intercept of -1, which matches with the equation y = -1/2x - 1.

Explanation

The equation for the line shown in the graph can be determined by comparing the slope and y-intercept values of the given equations with the slope and y-intercept values of the line in the graph. The line in the graph has a slope of -1/2 and a y-intercept of -1, which matches with the equation y = -1/2x - 1.

3) Determine the y-intercept of a line that passess through the points X(-2, -5) and Y(3, -4)

-1

-3

-23/5

-27/5

To determine the y-intercept of a line passing through two points, we can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.

First, we need to find the slope (m) of the line using the formula: m = (y2 - y1) / (x2 - x1). Substituting the coordinates of the two points, we get m = (-4 - (-5)) / (3 - (-2)) = 1/5.

Next, we can choose either point (X or Y) and substitute its coordinates along with the slope into the slope-intercept form to solve for b. Let's use point X(-2, -5): -5 = (1/5)(-2) + b. Solving for b, we get b = -23/5.

Therefore, the y-intercept of the line passing through the points X(-2, -5) and Y(3, -4) is -23/5.

Explanation

To determine the y-intercept of a line passing through two points, we can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.

First, we need to find the slope (m) of the line using the formula: m = (y2 - y1) / (x2 - x1). Substituting the coordinates of the two points, we get m = (-4 - (-5)) / (3 - (-2)) = 1/5.

Next, we can choose either point (X or Y) and substitute its coordinates along with the slope into the slope-intercept form to solve for b. Let's use point X(-2, -5): -5 = (1/5)(-2) + b. Solving for b, we get b = -23/5.

Therefore, the y-intercept of the line passing through the points X(-2, -5) and Y(3, -4) is -23/5.

4)

Karissa has been saving money on a monthly basis for two years. She started her account with an initial lump sum and deposits the same amount each month. The table shows her account balance without interest.

Determine an equation that represents the relationship.

B = 88m + 3

B = 45m + 175

B = 55m + 135

B = 60m + 115

The equation B = 45m + 175 represents the relationship between the account balance (B) and the number of months (m) that Karissa has been saving money. The coefficient of m (45) indicates that the account balance increases by 45 units each month. The constant term (175) represents the initial lump sum that Karissa started her account with. Therefore, this equation accurately represents the relationship described in the given scenario.

Explanation

The equation B = 45m + 175 represents the relationship between the account balance (B) and the number of months (m) that Karissa has been saving money. The coefficient of m (45) indicates that the account balance increases by 45 units each month. The constant term (175) represents the initial lump sum that Karissa started her account with. Therefore, this equation accurately represents the relationship described in the given scenario.

5)

Which is the equation of the line shown in the graph?

X = - 3

Y = - 3

Y = x - 3

Y = - x - 3

The equation of the line shown in the graph is y = -3. This can be determined by observing that the line is a horizontal line passing through the y-coordinate -3. The equation y = -3 represents a horizontal line at y = -3 on the graph.

Explanation

The equation of the line shown in the graph is y = -3. This can be determined by observing that the line is a horizontal line passing through the y-coordinate -3. The equation y = -3 represents a horizontal line at y = -3 on the graph.

6) If the equation y = 1/3x + b represents a line that passes through the point (5, -3), determine the value of the y-intercept

-14/3

4/3

4

6

The equation y = 1/3x + b represents a line in slope-intercept form, where the coefficient of x is the slope and b is the y-intercept. Since the line passes through the point (5, -3), we can substitute the coordinates into the equation to solve for b. Plugging in x = 5 and y = -3, we get -3 = 1/3(5) + b. Simplifying the equation, we have -3 = 5/3 + b. To isolate b, we subtract 5/3 from both sides, resulting in -14/3 = b. Therefore, the value of the y-intercept is -14/3.

Explanation

The equation y = 1/3x + b represents a line in slope-intercept form, where the coefficient of x is the slope and b is the y-intercept. Since the line passes through the point (5, -3), we can substitute the coordinates into the equation to solve for b. Plugging in x = 5 and y = -3, we get -3 = 1/3(5) + b. Simplifying the equation, we have -3 = 5/3 + b. To isolate b, we subtract 5/3 from both sides, resulting in -14/3 = b. Therefore, the value of the y-intercept is -14/3.

7) If the equation y = mx - 6 represents a line that passes through the point (-8, 5), determine the slope value, m.

- 11 / 8

-2 / 5

- 1 / 8

2 / 5

The equation y = mx - 6 represents a linear equation in slope-intercept form, where m is the slope of the line. To determine the slope value, we can compare the given equation with the standard form y = mx + b, where b is the y-intercept. In this case, the y-intercept is -6. Since the line passes through the point (-8, 5), we can substitute these values into the equation and solve for m. By substituting the values, we get 5 = (-11/8)(-8) - 6. Simplifying this equation, we find that m = -11/8. Therefore, the correct answer is -11/8.

Explanation

The equation y = mx - 6 represents a linear equation in slope-intercept form, where m is the slope of the line. To determine the slope value, we can compare the given equation with the standard form y = mx + b, where b is the y-intercept. In this case, the y-intercept is -6. Since the line passes through the point (-8, 5), we can substitute these values into the equation and solve for m. By substituting the values, we get 5 = (-11/8)(-8) - 6. Simplifying this equation, we find that m = -11/8. Therefore, the correct answer is -11/8.

8) If the equation y = -2x + b represents a line that passes through the point (4, 1), determine the value of the y-intercept

-9

-7

6

9

The equation y = -2x + b represents a line with a slope of -2. To find the y-intercept, we can substitute the coordinates of the given point (4, 1) into the equation and solve for b. Plugging in x = 4 and y = 1, we get 1 = -2(4) + b. Simplifying this equation, we have 1 = -8 + b. Adding 8 to both sides, we find that b = 9. Therefore, the value of the y-intercept is 9.

Explanation

The equation y = -2x + b represents a line with a slope of -2. To find the y-intercept, we can substitute the coordinates of the given point (4, 1) into the equation and solve for b. Plugging in x = 4 and y = 1, we get 1 = -2(4) + b. Simplifying this equation, we have 1 = -8 + b. Adding 8 to both sides, we find that b = 9. Therefore, the value of the y-intercept is 9.

9)

Determine which equation is the equation for the line shown in the graph

Y = -1/4 + 3

Y = 1/4 + 3

Y = -4x + 3

Y = 4x + 3

The equation for the line shown in the graph can be determined by observing the slope and y-intercept. The slope of the line is positive, indicating that it rises as x increases. The y-intercept is 3, which is the point where the line crosses the y-axis. The equation y = 4x + 3 is the only option that satisfies these conditions, as it has a positive slope of 4 and a y-intercept of 3.

Explanation

The equation for the line shown in the graph can be determined by observing the slope and y-intercept. The slope of the line is positive, indicating that it rises as x increases. The y-intercept is 3, which is the point where the line crosses the y-axis. The equation y = 4x + 3 is the only option that satisfies these conditions, as it has a positive slope of 4 and a y-intercept of 3.

10)

George works a job where he receives an hourly wage plus a bonus for being on time every day. The table below shows his wages for four weeks in which he was on time every day.

Write an equation to represent the relationship.

Y = 15x + 10

Y = 15.5x - 20

Y = 15.5x + 20

Y = 20x + 10

The equation y = 15.5x + 20 represents the relationship between George's wages (y) and the number of hours he worked (x). The coefficient of 15.5 represents the hourly wage, while the constant term of 20 represents the bonus for being on time every day. The equation shows that for every hour George works, he earns $15.5 as his hourly wage and receives an additional $20 as a bonus for being on time every day.

Explanation

The equation y = 15.5x + 20 represents the relationship between George's wages (y) and the number of hours he worked (x). The coefficient of 15.5 represents the hourly wage, while the constant term of 20 represents the bonus for being on time every day. The equation shows that for every hour George works, he earns $15.5 as his hourly wage and receives an additional $20 as a bonus for being on time every day.