Problem 46 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ x-3=11 $$

Step-by-Step Solution

Answer

The solution to the equation \(x - 3 = 11\) is \(x = 14\)

Step 1: Identify the Equation

The given equation to solve for \(x\) is \(x - 3 = 11\)

Step 2: Move Constant from the Left Side to the Right Side of the Equation

The next step is to get \(x\) alone on one side of the equation. We can do this by adding \(3\) to both sides of the equation to cancel out \(-3\) on the left-hand side which gives \(x = 11 + 3\)

Step 3: Simplify the Equation

The final step is to simplify right side of the equation, which results in \(x = 14\)

Key Concepts

Algebraic Equations

An algebraic equation is like a scale, where you want both sides to be balanced. In the given problem, you're presented with the equation
\[x - 3 = 11\].
Think of it as saying, 'Some number minus 3 is the same as 11.' Your task is to find that mysterious 'some number,' which we represent with the variable 'x.' To keep the balance when solving the equation, whatever you do to one side—like adding or subtracting—you must do to the other side as well. This keeps the equation equal, or 'in balance,' ensuring that the solution we find for 'x' is correct in the context of the math problem at hand.

Equation Simplification

Simplifying an equation is like cleaning up your room so that everything is easier to find. To simplify the equation
\[x - 3 = 11\],
you want to remove unnecessary items, making 'x' the centerpiece. Doing this can involve 'moving' numbers across the equals sign by carrying out the opposite operation; if there's a minus on one side, you use a plus on the other, and vice versa. In our equation, we add 3 to both sides, resulting in
\[x = 14\].
The equation is now as simple as it gets—no extra numbers or operations around our variable 'x.'

Isolating Variables

To isolate the variable means to get the variable 'x' alone on one side of the equation. Picture 'x' as someone who wants to be left alone on one side of a room (the equation). In the problem
\[x - 3 = 11\],
'x' isn't alone because there's a pesky '-3' crowding it. To give 'x' some space, we add 3 (the opposite of -3) to both sides. Now 'x' has the whole side to itself, leading to the simplified form
\[x = 14\].
Isolating the variable is a crucial step to solve an equation, as it provides the solution to what the variable represents.

Algebraic Expressions

An algebraic expression is a phrase in math that can contain numbers, variables, and operations, but it doesn’t have an equals sign—so it’s not an equation. In our problem, before solving the equation, the part
\[x - 3\]
is an expression since it doesn’t tell us anything equals that expression. It's essential to understand expressions because they're the building blocks of equations. When we solved
\[x - 3 = 11\],
'x - 3' and '11' were two expressions that we said were equal, and that's what transformed our expression into an equation. Knowing how to manipulate expressions is key to solving equations.