Solve For The Variable In The Brackets.a) A 4 + A = 5 \frac{a}{4+a}=5 4 + A A ​ = 5 (Solve For A A A )b) F = G M M R 2 F=\frac{G M M}{r^2} F = R 2 G M M ​ (Solve For R R R )

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In mathematics, solving for variables in equations is a fundamental concept that involves isolating the variable on one side of the equation. This is a crucial skill in algebra and is used extensively in various mathematical disciplines, including physics and engineering. In this article, we will explore two examples of solving for variables in equations: one from algebra and the other from physics.

Example 1: Solving for aa in the Equation a4+a=5\frac{a}{4+a}=5

The given equation is a4+a=5\frac{a}{4+a}=5. To solve for aa, we need to isolate aa on one side of the equation. We can start by multiplying both sides of the equation by 4+a4+a, which is the denominator of the fraction on the left-hand side.

\frac{a}{4+a} = 5
\Rightarrow a = 5(4+a)

Next, we can distribute the 55 on the right-hand side of the equation to get:

a = 20 + 5a

Now, we can subtract 5a5a from both sides of the equation to get:

-4a = 20

Finally, we can divide both sides of the equation by −4-4 to solve for aa:

a = -5

Therefore, the value of aa that satisfies the equation a4+a=5\frac{a}{4+a}=5 is a=−5a=-5.

Example 2: Solving for rr in the Equation f=GmMr2f=\frac{G m M}{r^2}

The given equation is f=GmMr2f=\frac{G m M}{r^2}. To solve for rr, we need to isolate rr on one side of the equation. We can start by multiplying both sides of the equation by r2r^2, which is the denominator of the fraction on the right-hand side.

f = \frac{G m M}{r^2}
\Rightarrow f r^2 = G m M

Next, we can divide both sides of the equation by ff to get:

r^2 = \frac{G m M}{f}

Finally, we can take the square root of both sides of the equation to solve for rr:

r = \sqrt{\frac{G m M}{f}}

Therefore, the value of rr that satisfies the equation f=GmMr2f=\frac{G m M}{r^2} is r=GmMfr=\sqrt{\frac{G m M}{f}}.

Conclusion

Solving for variables in equations is a fundamental concept in mathematics that involves isolating the variable on one side of the equation. In this article, we have explored two examples of solving for variables in equations: one from algebra and the other from physics. By following the steps outlined in this article, readers should be able to solve for variables in equations with ease.

Tips and Tricks

  • When solving for variables in equations, it is essential to isolate the variable on one side of the equation.
  • Use inverse operations to eliminate the variable from the equation.
  • Simplify the equation by combining like terms.
  • Check the solution by plugging it back into the original equation.

Real-World Applications

Solving for variables in equations has numerous real-world applications in various fields, including:

  • Physics: Solving for variables in equations is essential in physics to describe the motion of objects, the behavior of particles, and the properties of materials.
  • Engineering: Solving for variables in equations is crucial in engineering to design and optimize systems, structures, and processes.
  • Economics: Solving for variables in equations is essential in economics to model economic systems, predict economic trends, and make informed decisions.

Common Mistakes to Avoid

  • Failing to isolate the variable on one side of the equation.
  • Using the wrong inverse operation to eliminate the variable.
  • Failing to simplify the equation by combining like terms.
  • Not checking the solution by plugging it back into the original equation.

Conclusion

In our previous article, we explored two examples of solving for variables in equations: one from algebra and the other from physics. In this article, we will answer some frequently asked questions about solving for variables in equations.

Q: What is the first step in solving for a variable in an equation?

A: The first step in solving for a variable in an equation is to isolate the variable on one side of the equation. This can be done by using inverse operations, such as addition, subtraction, multiplication, or division, to eliminate the variable from the other side of the equation.

Q: How do I know which inverse operation to use?

A: To determine which inverse operation to use, you need to look at the operation that is being performed on the variable. For example, if the variable is being multiplied by a number, you can use division to eliminate the variable. If the variable is being added to a number, you can use subtraction to eliminate the variable.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation 2x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, you can simplify the expression and solve for x.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation, on the other hand, is a single equation that is solved independently.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use substitution or elimination. Substitution involves solving one equation for one variable and then plugging that value into the other equation. Elimination involves adding or subtracting the equations to eliminate one variable.

Q: What is the difference between a linear system and a nonlinear system?

A: A linear system is a system of equations in which the highest power of the variable is 1. A nonlinear system, on the other hand, is a system of equations in which the highest power of the variable is greater than 1.

Q: How do I solve a nonlinear system?

A: To solve a nonlinear system, you can use numerical methods, such as the Newton-Raphson method, or graphical methods, such as the graphing calculator.

Conclusion

Solving for variables in equations is a fundamental concept in mathematics that involves isolating the variable on one side of the equation. By following the steps outlined in this article, readers should be able to solve for variables in equations with ease. Remember to isolate the variable, use inverse operations, simplify the equation, and check the solution to ensure accuracy.

Tips and Tricks

  • Use inverse operations to eliminate the variable from the equation.
  • Simplify the equation by combining like terms.
  • Check the solution by plugging it back into the original equation.
  • Use the quadratic formula to solve quadratic equations.
  • Use substitution or elimination to solve systems of equations.

Real-World Applications

Solving for variables in equations has numerous real-world applications in various fields, including:

  • Physics: Solving for variables in equations is essential in physics to describe the motion of objects, the behavior of particles, and the properties of materials.
  • Engineering: Solving for variables in equations is crucial in engineering to design and optimize systems, structures, and processes.
  • Economics: Solving for variables in equations is essential in economics to model economic systems, predict economic trends, and make informed decisions.

Common Mistakes to Avoid

  • Failing to isolate the variable on one side of the equation.
  • Using the wrong inverse operation to eliminate the variable.
  • Failing to simplify the equation by combining like terms.
  • Not checking the solution by plugging it back into the original equation.