Problem 36 - Algebra 1

Question

MENTAL MATH Write a question that could be used to solve the equation. Then use mental math to solve the equation. $$5 q-2=3$$

Step-by-Step Solution

Answer

The value for q that satisfies the equation is q = 1

Step 1: Formulate the question

In this step you should formulate the question that leads to the resolution of the equation. A possible question could be: 'What is the value for q that satisfies the equation \(5q - 2 = 3\)?'

Step 2: Rearrange the equation

In this step, isolate the variable q on one side of the equation. You can do this by adding 2 to both sides of the equation. This gives us the new equation: \(5q = 3 + 2\)

Step 3: Simplify the equation

Simplify the right side of the equation by performing the operation. This gives us: \(5q = 5\)

Step 4: Solve for q

Finally, solve for q by dividing both sides of the equation by 5. Doing so gives us: \(q = 5 / 5\)

Step 5: Simplify the final solution

Simplify the right side of the equation to find the value of q: \(q = 1\)

Key Concepts

Mental Math

Mental math is all about using your brain to solve numerical problems without the aid of calculators or even paper and pencil. It's a valuable skill that helps not only in academic settings but also in everyday life when quick calculations are needed. In our example, the equation was \(5q - 2 = 3\). Using mental math, you would first recognize that by adding 2 to both sides, you balance out the equation, leading to \(5q = 5\). Understanding basic arithmetic operations allows you to see that if 5 times a certain number equals 5, that number must be 1. Thus, without writing anything down, you'd quickly deduce that \(q = 1\).

To improve one's mental math abilities, some tips include practicing with times tables, breaking numbers down into smaller parts to make them easier to manage, and regularly challenging oneself with mental calculations. Strengthening these skills can make solving equations in your head a much swifter process.

Isolating Variables

The process of isolating variables is crucial when it comes to solving equations. This means manipulating the equation so that the variable you're solving for is on one side of the equation by itself. In the given exercise, we are aiming to find the value of the variable \(q\). Starting with \(5q - 2 = 3\), you add 2 to both sides, effectively 'moving' the -2 over to the other side. It's like balancing scales: what you do to one side, you do to the other to keep it balanced.

To isolate variables effectively, remember the inverse operations: addition/subtraction are the inverse of each other, as are multiplication/division. Use these operations to 'undo' what's being done to the variable. Once the variable is isolated, like in our equation where we ended up with \(5q = 5\), it's a straightforward step to find its value.

Simplifying Equations

Simplifying equations is an essential step in finding their solutions. It involves reducing the complexity of the equation by combining like terms, eliminating fractions, or carrying out basic arithmetic operations. In our solution, after isolating the variable \(q\), we ended up with the equation \(5q = 5\). The simplification here was straightforward, involving just one division step, but the process could be more complex with different equations.

Remember, the goal of simplification is to make the equation as 'clean' and manageable as possible before solving for the variable. This means doing things like combining terms (e.g., \(2q + 3q = 5q\)) or reducing fractions. It's a bit like tidying up a room so you can easily find what you're looking for. After the equation has been simplified to \(q = 1\), we've reached the solution in the cleanest, simplest form possible.