Problem 39 - Algebra 1

Question

Write a question that can be used to solve the equation. Then use mental math to solve the equation. \(x+17=25\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 8\).

Step 1: Identify the Equation

The equation to be solved is \(x + 17 = 25\). The aim is to isolate 'x'. This requires using inverse operations to cancel out '+17' on one side.

Step 2: Use Inverse Operation

The inverse operation of addition is subtraction. To eliminate '+17' on the left side, subtract '17' from both sides of the equation, which results in \(x + 17 - 17 = 25 - 17\).

Step 3: Simplify the Equation

On simplifying both the sides of the equation, it changes to \(x = 25 - 17\). Calculating the right side of the equation gives the solution \(x = 8\).

Key Concepts

Inverse Operations

When solving linear equations, inverse operations are tools that help us simplify the equation and isolate the variable. In our equation, we started with \(x + 17 = 25\). Inverse operations involve doing the opposite mathematical operation to cancel out terms and simplify equations.
In this case, because we are dealing with addition on the left side of the equation, we apply subtraction to both sides. This is because subtraction is the inverse operation of addition.

• To cancel out '+17', subtract '17' from both sides.
• Ensure the equation remains balanced by applying the operation equally to both sides.

Using inverse operations is vital. It allows us to systematically remove the unwanted numbers attached to the variable, bringing us closer to revealing the value of 'x'.

Mental Math

Mental math is a powerful skill that makes solving equations quicker and easier, especially for simple linear equations. It involves calculating the values without writing them down.
In the equation \(x + 17 = 25\), once we simplify it by removing +17 through subtraction, we get \(x = 25 - 17\). Rather than using a calculator or doing paper-pencil calculations, we can quickly determine that the result is \(8\) through mental math.

• Focus on subtracting numbers mentally by breaking them into simpler parts if needed.
• In this example, start by subtracting 20 from 25 to get 5, then add 3 to reach 8.
• Practicing mental math enhances numerical fluency which saves time in exams or real-life problem-solving situations.

Using mental math for such calculations can speed up the process and build confidence in handling numbers.

Isolating Variables

Isolating the variable is the objective of solving equations as it allows us to find the value of the unknown variable. Our goal in this process is to have \(x\) by itself on one side of the equation. Starting with \(x + 17 = 25\), the ultimate target is to rearrange the equation such that only \(x\) remains on one side.
The steps involved mean removing any numbers or terms added, multiplied, subtracted, or divided around \(x\) using appropriate inverse operations. After subtracting 17 from both sides, the equation simplifies to \(x = 8\).

• Isolating variables hinges on performing opposite mathematical operations.
• Ensure every step keeps the equation balanced on both sides.
• Always recheck your work by substituting the found value back into the original equation to confirm it's correctly solved.

Finding \(x\) successfully means understanding each operation's role and applying it consistently to maintain an equilibrium within the equation.