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.
Subtract 5 from both sides:
$2q + 5 – 5 = 15 – 5$
Simplifies to:
$2q = 10$
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 $c = 4$
Calculate the discriminant:
$b^2 – 4ac = (-4)^2 – 4(1)(4) = 16 – 16 = 0$
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:
$begin{cases}
2q + r = 10
3q – r = 5
end{cases}$
Add the equations to eliminate $r$:
$(2q + r) + (3q – r) = 10 + 5$
Simplifies to:
$5q = 15$
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 $q$