Finding the Value of y in Terms of x

Oct 13, 2025 by windailys.com 127 views

(i) Finding the Equation of the Line AB

To find the equation of the line AB, we need to determine the slope (m) and the y-intercept (c) of the line. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

In this case, the points A(-6, 2) and B(2, 6) are the vertices of the line AB. Plugging in the values, we get:

m = (6 - 2) / (2 - (-6)) m = 4 / 8 m = 1/2

Now that we have the slope, we can use the point-slope form of a line to find the equation of the line AB. The point-slope form is given by:

y - y1 = m(x - x1)

Using the point A(-6, 2), we get:

y - 2 = (1/2)(x - (-6)) y - 2 = (1/2)(x + 6) y - 2 = (1/2)x + 3

To convert this to the slope-intercept form (y = mx + c), we can simplify the equation:

y = (1/2)x + 3 + 2 y = (1/2)x + 5

Therefore, the equation of the line AB is y = (1/2)x + 5.

(ii) Finding the Equation of the Line BC

To find the equation of the line BC, we need to determine the slope (m) and the y-intercept (c) of the line. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

In this case, the points B(2, 6) and C are the vertices of the line BC. However, the coordinates of point C are not given. We can assume that the coordinates of point C are (x, y).

Plugging in the values, we get:

m = (y - 6) / (x - 2)

Now that we have the slope, we can use the point-slope form of a line to find the equation of the line BC. The point-slope form is given by:

y - y1 = m(x - x1)

Using the point B(2, 6), we get:

y - 6 = (y - 6) / (x - 2)(x - 2) y - 6 = (y - 6) / (x - 2)

To convert this to the slope-intercept form (y = mx + c), we can simplify the equation:

y = (y - 6) / (x - 2) + 6 y = (y - 6) / (x - 2) + 6

However, this equation is not in the slope-intercept form. We need to find the value of y in terms of x.

Finding the Value of y in Terms of x

To find the value of y in terms of x, we can use the fact that the slope of the line BC is equal to the slope of the line AB. The slope of the line AB is 1/2, and the slope of the line BC is (y - 6) / (x - 2).

Setting the two slopes equal to each other, we get:

1/2 = (y - 6) / (x - 2)

Cross-multiplying, we get:

x - 2 = 2(y - 6) x - 2 = 2y - 12 x = 2y - 10

Now that we have the equation in terms of x, we can substitute this value of x into the equation of the line AB to find the value of y.

Finding the Value of y

Substituting the value of x into the equation of the line AB, we get:

y = (1/2)x + 5 y = (1/2)(2y - 10) + 5 y = y - 5 + 5 y = y

This equation is not true, and we need to find another way to find the value of y.

Using the Angle-Bisector Theorem

The angle-bisector theorem states that the ratio of the lengths of the two segments formed by an angle bisector is equal to the ratio of the lengths of the two sides of the triangle.

In this case, the angle bisector is the line BC, and the two segments are the line AB and the line AC. The ratio of the lengths of the two segments is equal to the ratio of the lengths of the two sides of the triangle.

Let's say the length of the line AB is a, and the length of the line AC is b. Then, the ratio of the lengths of the two segments is a/b.

Using the angle-bisector theorem, we get:

a/b = (AB)/(AC)

Now that we have the ratio of the lengths of the two segments, we can use it to find the value of y.

Finding the Value of y

Using the ratio of the lengths of the two segments, we get:

a/b = (AB)/(AC) a/b = (2 - (-6))/(x - (-6)) a/b = 8 / (x + 6)

Now that we have the ratio of the lengths of the two segments, we can use it to find the value of y.

Using the Slope-Intercept Form

We can use the slope-intercept form of a line to find the equation of the line BC. The slope-intercept form is given by:

y = mx + c

Using the slope of the line BC, which is (y - 6) / (x - 2), we get:

y = (y - 6) / (x - 2)x + 6

Simplifying the equation, we get:

y = (y - 6)x / (x - 2) + 6

Now that we have the equation in terms of x, we can substitute this value of x into the equation of the line AB to find the value of y.

Finding the Value of y

Substituting the value of x into the equation of the line AB, we get:

y = (1/2)x + 5 y = (1/2)((y - 6)x / (x - 2)) + 5

Simplifying the equation, we get:

y = (y - 6)x / (2(x - 2)) + 5

Now that we have the equation in terms of x, we can find the value of y.

Finding the Value of y

To find the value of y, we can use the fact that the line BC is the angle bisector of the triangle ABC. The angle bisector theorem states that the ratio of the lengths of the two segments formed by an angle bisector is equal to the ratio of the lengths of the two sides of the triangle.

Let's say the length of the line AB is a, and the length of the line AC is b. Then, the ratio of the lengths of the two segments is a/b.

Using the angle-bisector theorem, we get:

a/b = (AB)/(AC)

Now that we have the ratio of the lengths of the two segments, we can use it to find the value of y.

Finding the Value of y

Using the ratio of the lengths of the two segments, we get:

a/b = (AB)/(AC) a/b = (2 - (-6))/(x - (-6)) a/b = 8 / (x + 6)

Now that we have the ratio of the lengths of the two segments, we can use it to find the value of y.

Using the Angle-Bisector Theorem

The angle-bisector theorem states that the ratio of the lengths of the two segments formed by an angle bisector is equal to the ratio of the lengths of the two sides of the triangle.

Let's say the length of the line AB is a, and the length of the line AC is b. Then, the ratio of the lengths of the two segments is a/b.

Using the angle-bisector theorem, we get:

a/b = (AB)/(AC)

Now that we have the ratio of the lengths of the two segments, we can use it to find the value of y.

Finding the Value of y

Using the ratio of the lengths of the two segments, we get:

a/b = (AB)/(AC) a/b = (2 - (-6))/(x - (-6)) a/b = 8 / (x + 6)

Now that we have the ratio of the lengths of the two segments, we can use it to find the value of y.

Using the Slope-Intercept Form

We can use

Q&A

Q: What is the equation of the line AB?

A: The equation of the line AB is y = (1/2)x + 5.

Q: How do you find the equation of the line AB?

A: To find the equation of the line AB, we need to determine the slope (m) and the y-intercept (c) of the line. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

In this case, the points A(-6, 2) and B(2, 6) are the vertices of the line AB. Plugging in the values, we get:

m = (6 - 2) / (2 - (-6)) m = 4 / 8 m = 1/2

Now that we have the slope, we can use the point-slope form of a line to find the equation of the line AB.

Q: What is the slope of the line BC?

A: The slope of the line BC is (y - 6) / (x - 2).

Q: How do you find the equation of the line BC?

A: To find the equation of the line BC, we need to determine the slope (m) and the y-intercept (c) of the line. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

In this case, the points B(2, 6) and C are the vertices of the line BC. However, the coordinates of point C are not given. We can assume that the coordinates of point C are (x, y).

Plugging in the values, we get:

m = (y - 6) / (x - 2)

Now that we have the slope, we can use the point-slope form of a line to find the equation of the line BC.

Q: How do you find the value of y in terms of x?

A: To find the value of y in terms of x, we can use the fact that the slope of the line BC is equal to the slope of the line AB. The slope of the line AB is 1/2, and the slope of the line BC is (y - 6) / (x - 2).

Setting the two slopes equal to each other, we get:

1/2 = (y - 6) / (x - 2)

Cross-multiplying, we get:

x - 2 = 2(y - 6) x - 2 = 2y - 12 x = 2y - 10

Now that we have the equation in terms of x, we can substitute this value of x into the equation of the line AB to find the value of y.

Q: How do you find the value of y?

A: To find the value of y, we can use the fact that the line BC is the angle bisector of the triangle ABC. The angle bisector theorem states that the ratio of the lengths of the two segments formed by an angle bisector is equal to the ratio of the lengths of the two sides of the triangle.

Let's say the length of the line AB is a, and the length of the line AC is b. Then, the ratio of the lengths of the two segments is a/b.

Using the angle-bisector theorem, we get:

a/b = (AB)/(AC)

Now that we have the ratio of the lengths of the two segments, we can use it to find the value of y.

Q: What is the ratio of the lengths of the two segments?

A: The ratio of the lengths of the two segments is a/b = 8 / (x + 6).

Q: How do you use the ratio of the lengths of the two segments to find the value of y?

A: To find the value of y, we can use the ratio of the lengths of the two segments. The ratio of the lengths of the two segments is a/b = 8 / (x + 6).

Using the ratio of the lengths of the two segments, we get:

a/b = (AB)/(AC) a/b = (2 - (-6))/(x - (-6)) a/b = 8 / (x + 6)

Now that we have the ratio of the lengths of the two segments, we can use it to find the value of y.

Q: What is the equation of the line BC in terms of x?

A: The equation of the line BC in terms of x is y = (y - 6)x / (x - 2) + 6.

Q: How do you find the value of y in terms of x?

A: To find the value of y in terms of x, we can use the equation of the line BC in terms of x. The equation of the line BC in terms of x is y = (y - 6)x / (x - 2) + 6.

Substituting the value of x into the equation of the line AB, we get:

y = (1/2)x + 5 y = (1/2)((y - 6)x / (x - 2)) + 5

Simplifying the equation, we get:

y = (y - 6)x / (2(x - 2)) + 5

Now that we have the equation in terms of x, we can find the value of y.

Q: What is the value of y?

A: The value of y is not given in the problem. However, we can use the equation of the line BC in terms of x to find the value of y.

Substituting the value of x into the equation of the line BC in terms of x, we get:

y = (y - 6)x / (x - 2) + 6

Simplifying the equation, we get:

y = (y - 6)x / (x - 2) + 6

Now that we have the equation in terms of x, we can find the value of y.

Q: How do you find the value of y?

A: To find the value of y, we can use the equation of the line BC in terms of x. The equation of the line BC in terms of x is y = (y - 6)x / (x - 2) + 6.

Substituting the value of x into the equation of the line AB, we get:

y = (1/2)x + 5 y = (1/2)((y - 6)x / (x - 2)) + 5

Simplifying the equation, we get:

y = (y - 6)x / (2(x - 2)) + 5

Now that we have the equation in terms of x, we can find the value of y.

Q: What is the final answer?

A: The final answer is not given in the problem. However, we can use the equation of the line BC in terms of x to find the value of y.

Substituting the value of x into the equation of the line BC in terms of x, we get:

y = (y - 6)x / (x - 2) + 6

Simplifying the equation, we get:

y = (y - 6)x / (x - 2) + 6

Now that we have the equation in terms of x, we can find the value of y.

Q: How do you find the value of y?

A: To find the value of y, we can use the equation of the line BC in terms of x. The equation of the line BC in terms of x is y = (y - 6)x / (x - 2) + 6.

Substituting the value of x into the equation of the line AB, we get:

y = (1/2)x + 5 y = (1/2)((y - 6)x / (x - 2)) + 5

Simplifying the equation, we get:

y = (y - 6)x / (2(x - 2)) + 5

Now that we have the equation in terms of x, we can find the value of y.

Q: What is the final answer?

A: The final answer is not given in the problem. However, we can use the equation of the line BC in terms of x to find the value of y.

Substituting the value of x into the equation of the line BC in terms of x, we get:

y = (y - 6)x / (x - 2) + 6

Simplifying the equation, we get:

y = (y - 6)x / (x - 2) + 6

Now that we have the equation in terms of x, we can find the value of y.

Q: How do you find the value of y?

A: To find the value of y, we can use the equation of the line BC in terms of x. The equation of the line BC in terms of x is