How to Solve for q in an Equation?

Solving for a variable in an equation is a fundamental skill in algebra. Let’s break down the process of solving for $q$ in different types of equations.

Basic Linear Equations

In a simple linear equation like $2q + 5 = 15$, you can solve for $q$ by isolating it on one side of the equation.

1. Subtract 5 from both sides:

   $2q + 5 – 5 = 15 – 5$

   Simplifies to:

   $2q = 10$

2. Divide both sides by 2:

   $frac{2q}{2} = frac{10}{2}$

   Simplifies to:

   $q = 5$

Quadratic Equations

For quadratic equations like $q^2 – 4q + 4 = 0$, you can solve for $q$ using the quadratic formula:

$q = frac{-b pm sqrt{b^2 – 4ac}}{2a}$

In this case, $a = 1$, $b = -4$, and $c = 4$

1. Calculate the discriminant:

   $b^2 – 4ac = (-4)^2 – 4(1)(4) = 16 – 16 = 0$

2. Apply the quadratic formula:

   $ q = frac{-(-4) pm sqrt{0}}{2(1)} = frac{4 pm 0}{2} = 2$

So, $q = 2$ is the solution.

Systems of Equations

Sometimes, you might need to solve for $q$ in a system of equations. For example:

$begin{cases}
2q + r = 10
3q – r = 5
end{cases}$

1. Add the equations to eliminate $r$:

   $(2q + r) + (3q – r) = 10 + 5$

   Simplifies to:

   $5q = 15$

2. Divide both sides by 5:

   $frac{5q}{5} = frac{15}{5}$

   Simplifies to:

   $q = 3$

Conclusion

Solving for $q$ involves isolating the variable through various algebraic techniques, depending on the type of equation. Whether it’s a simple linear equation, a quadratic equation, or a system of equations, the key is to follow systematic steps to isolate $q$

Citations

1. 1. Khan Academy – Solving Equations
2. 2. Math is Fun – Solving Equations
3. 3. Purplemath – Solving Quadratic Equations