Question

Solve \(-3 x+17=5\)

Short answer

x = 4

Step-by-step solution

Write down the equation

Start with the given equation: \[ -3x + 17 = 5 \]

Isolate the term with the variable

Subtract 17 from both sides to isolate the term with the variable. \[ -3x + 17 - 17 = 5 - 17 \] This simplifies to: \[ -3x = -12 \]

Solve for x

Divide both sides by -3 to solve for \(x\). \[ x = \frac{-12}{-3} \] This simplifies to: \[ x = 4 \]

Key concepts

Solving Equations

Solving equations might seem tricky, but it's quite manageable with practice! An equation like \( -3x + 17 = 5 \) is asking what value of \( x \) makes the statement true. Here are the basic steps:
First, understand that our goal is to find the value of the variable (here, \( x \)) that satisfies the equation. We approach this by performing operations that simplify the equation until \( x \) is isolated.
When we solve equations, we often use operations like addition, subtraction, multiplication, and division. Each step we take should keep the equation balanced. This means whatever operation you do to one side, you must do to the other side too!
Let's break down the example presented:
The equation is: \[ -3x + 17 = 5 \]
We aim to isolate \( x \) on one side of the equation, making it easier to see its value.

Isolating Variables

Isolating variables means getting the variable by itself on one side of the equation. This often means moving numbers or other variables to the other side. Here's how it works with our example:
Start with the equation: \[ -3x + 17 = 5 \]
To isolate \( x \), we need to move the 17. We do this by subtracting 17 from both sides. Why subtract? Because the opposite of addition (here, +17) is subtraction. This cancels out the 17 on the left side, making the equation simpler.
So, subtract 17 from both sides: \[ -3x + 17 - 17 = 5 - 17 \]
Now, simplify both sides: \[ -3x = -12 \]
See how \( x \) is becoming isolated? We're almost done!

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to simplify expressions and solve equations. It's like a clever trick to make complex equations easier to handle. Let's wrap up the earlier example:
We ended at: \[ -3x = -12 \]
Now, divide both sides by -3 to completely isolate \( x \). Dividing is the opposite of multiplying (here, -3 times \( x \)).
Dividing both sides gives: \[ x = \frac{-12}{-3} \]
Simplify the division: \[ x = 4 \]
And there we have it! Through algebraic manipulation, we've solved the equation and found that \( x = 4 \). Remember, every step in manipulating the equation keeps it balanced. This is the magic behind algebra – making changes while keeping things fair on both sides.
Everyday exercises like this help build a strong foundation, making algebra a powerful tool for solving real-world problems.