The Distance Between The Points \[$(10, Y)\$\] And \[$(4,8)\$\] Is \[$\sqrt{61}\$\].Arrange The Equations Below To Show The Process For Finding The Value Of \[$y\$\].$\[ \begin{array}{l} 1. \quad
Introduction
In mathematics, the distance between two points is a fundamental concept that is used in various fields, including geometry, trigonometry, and physics. The distance between two points can be calculated using the distance formula, which is derived from the Pythagorean theorem. In this article, we will explore how to find the distance between two points and use it to determine the value of y in a given equation.
The Distance Formula
The distance formula is used to calculate the distance between two points (x1, y1) and (x2, y2) in a coordinate plane. The formula is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where d is the distance between the two points.
Given Information
We are given two points: (10, y) and (4, 8). The distance between these two points is √61. We need to find the value of y.
Step 1: Substitute the Given Values into the Distance Formula
We will substitute the given values into the distance formula:
√61 = √((4 - 10)^2 + (8 - y)^2)
Step 2: Simplify the Equation
We will simplify the equation by evaluating the expressions inside the parentheses:
√61 = √((-6)^2 + (8 - y)^2)
Step 3: Expand the Squared Terms
We will expand the squared terms:
√61 = √(36 + (8 - y)^2)
Step 4: Simplify the Equation Further
We will simplify the equation further by evaluating the expression inside the parentheses:
√61 = √(36 + 64 - 16y + y^2)
Step 5: Combine Like Terms
We will combine like terms:
√61 = √(100 - 16y + y^2)
Step 6: Square Both Sides of the Equation
We will square both sides of the equation to eliminate the square root:
61 = 100 - 16y + y^2
Step 7: Rearrange the Equation
We will rearrange the equation to form a quadratic equation:
y^2 - 16y + 39 = 0
Step 8: Solve the Quadratic Equation
We will solve the quadratic equation using the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = -16, and c = 39.
Step 9: Evaluate the Quadratic Formula
We will evaluate the quadratic formula:
y = (16 ± √((-16)^2 - 4(1)(39))) / 2(1)
Step 10: Simplify the Expression
We will simplify the expression:
y = (16 ± √(256 - 156)) / 2
Step 11: Evaluate the Expression Inside the Square Root
We will evaluate the expression inside the square root:
y = (16 ± √100) / 2
Step 12: Simplify the Expression Further
We will simplify the expression further:
y = (16 ± 10) / 2
Step 13: Evaluate the Two Possible Solutions
We will evaluate the two possible solutions:
y = (16 + 10) / 2 or y = (16 - 10) / 2
Step 14: Simplify the Two Possible Solutions
We will simplify the two possible solutions:
y = 26 / 2 or y = 6 / 2
Step 15: Evaluate the Two Possible Solutions
We will evaluate the two possible solutions:
y = 13 or y = 3
Conclusion
In this article, we used the distance formula to find the value of y in a given equation. We started with the distance formula and substituted the given values into the equation. We then simplified the equation and solved for y using the quadratic formula. We evaluated the two possible solutions and found that y = 13 or y = 3.
Introduction
In our previous article, we explored how to find the distance between two points and use it to determine the value of y in a given equation. We used the distance formula and the quadratic formula to solve for y. In this article, we will answer some frequently asked questions about the distance between two points and finding the value of y.
Q: What is the distance formula?
A: The distance formula is a mathematical formula used to calculate the distance between two points (x1, y1) and (x2, y2) in a coordinate plane. The formula is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Q: How do I use the distance formula to find the distance between two points?
A: To use the distance formula, you need to substitute the coordinates of the two points into the formula and simplify the expression. For example, if you want to find the distance between the points (3, 4) and (6, 8), you would substitute x1 = 3, y1 = 4, x2 = 6, and y2 = 8 into the formula.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form ax^2 + bx + c = 0. The formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula to solve for y?
A: To use the quadratic formula, you need to substitute the values of a, b, and c into the formula and simplify the expression. For example, if you want to solve the quadratic equation y^2 - 16y + 39 = 0, you would substitute a = 1, b = -16, and c = 39 into the formula.
Q: What is the difference between the two possible solutions for y?
A: The two possible solutions for y are the values of y that satisfy the quadratic equation. In the case of the equation y^2 - 16y + 39 = 0, the two possible solutions are y = 13 and y = 3. These values of y are the solutions to the equation.
Q: How do I determine which solution is correct?
A: To determine which solution is correct, you need to substitute the values of y back into the original equation and check if the equation is true. In the case of the equation y^2 - 16y + 39 = 0, you would substitute y = 13 and y = 3 back into the equation and check if the equation is true.
Q: What if I get two different solutions for y?
A: If you get two different solutions for y, it means that the equation has two possible solutions. In this case, you need to check which solution is correct by substituting the values of y back into the original equation.
Q: Can I use the distance formula to find the distance between two points in 3D space?
A: Yes, you can use the distance formula to find the distance between two points in 3D space. The formula is:
d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
where x1, y1, and z1 are the coordinates of the first point, and x2, y2, and z2 are the coordinates of the second point.
Q: Can I use the quadratic formula to solve for y in a quadratic equation with complex coefficients?
A: Yes, you can use the quadratic formula to solve for y in a quadratic equation with complex coefficients. The formula is:
y = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
Conclusion
In this article, we answered some frequently asked questions about the distance between two points and finding the value of y. We covered topics such as the distance formula, the quadratic formula, and how to determine which solution is correct. We also discussed how to use the distance formula to find the distance between two points in 3D space and how to use the quadratic formula to solve for y in a quadratic equation with complex coefficients.