Equations with just one unknown can be like little mysteries. Let's learn how to solve them step by step.
A linear equation in one variable is an equation where you have one unknown, often represented by a letter like \(x\). Solving it means finding the value of this unknown that makes the equation true. Imagine you're trying to find out how many apples you have if you know the total weight and the weight of each apple. This is similar to solving a linear equation; you're figuring out the missing piece. Linear equations are useful because they help us understand and predict patterns in everyday situations, like budgeting money or calculating travel time. As you progress in maths, you'll encounter more complex equations, but mastering these simple ones is your first step.
### Worked example 1: Solving \(x + 3 = 7\)
Let's start gently.
**Problem:** Solve the equation \(x + 3 = 7\).
**Step 1:** Our goal is to find the value of \(x\). We start by undoing the addition of 3. To do this, we subtract 3 from both sides of the equation. This keeps the equation balanced, just like keeping a scale balanced by adding or removing the same weight from both sides.
\[x + 3 - 3 = 7 - 3\]
**Step 2:** After subtracting 3 from both sides, we simplify to find \(x\).
\[x = 4\]
**Answer:** The solution is \(x = 4\).
### Worked example 2: Solving \(2x - 5 = 9\)
Now a trickier one — same idea.
**Problem:** Solve the equation \(2x - 5 = 9\).
**Step 1:** First, we want to isolate the term with \(x\) by getting rid of the -5. We do this by adding 5 to both sides of the equation.
\[2x - 5 + 5 = 9 + 5\]
**Step 2:** Now we simplify both sides. This gives us a simpler equation where \(2x\) is isolated.
\[2x = 14\]
**Step 3:** To find \(x\), we divide both sides by 2. This step isolates \(x\) completely.
\[x = \frac{14}{2}\]
**Step 4:** Simplifying the division gives us the value of \(x\).
\[x = 7\]
**Answer:** The solution is \(x = 7\).
### Watch out
- Forgetting to perform the same operation on both sides of the equation. Always remember: what you do to one side, do to the other.
- Mistaking subtraction for addition or vice versa. Carefully check the operation needed to isolate the variable.
- Rushing through simplification steps and making arithmetic errors. Take your time to ensure each step is correct.
Your turn: Solve the equation \(3x + 4 = 19\).
Show your working — you can type it or snap a photo of it.