Problem 105 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ -3 x+8=-7 $$

Step-by-Step Solution

Answer

The solution is \( x = 5 \).

Step 1: Identify the Equation

Identify the equation given in the problem: \[ -3x + 8 = -7 \] Our goal is to find the value of \(x\) that satisfies this equation.

Step 2: Isolate the Variable Term

Subtract 8 from both sides of the equation to isolate the term with the variable \(x\):\[ -3x + 8 - 8 = -7 - 8 \]This simplifies to:\[ -3x = -15 \]

Step 3: Solve for the Variable

Divide both sides of the equation by -3 to solve for \(x\):\[ x = \frac{-15}{-3} \]Simplifying the division gives:\[ x = 5 \]

Step 4: Verify the Solution

Substitute \(x = 5\) back into the original equation to verify:\[ -3(5) + 8 = -7 \]This simplifies to:\[ -15 + 8 = -7 \]Since \(-7 = -7\), our solution is correct.

Key Concepts

Isolating the Variable

To solve an equation, start by isolating the variable, which means getting the unknown variable by itself on one side of the equation. In the equation \[-3x + 8 = -7\], we want to isolate \(x\). This involves eliminating any constants or coefficients attached to \(x\).
The first step is to remove the constant 8 from the left side. We do this by subtracting 8 from both sides of the equation. Remember, whatever operation you perform on one side of the equation, you must perform on the other to maintain balance. After this step, the equation becomes: \[-3x = -15\].
Now, the equation has only the term with the variable \(x\) on the left side. Isolating the variable is a critical step, as it simplifies the equation and makes it easier to find the solution.

Solving Equations

Once you've isolated the variable, the next step is to solve the equation for that variable. For the equation \[-3x = -15\], we must solve for \(x\) by removing the coefficient -3. This involves dividing both sides of the equation by -3.

• Perform the division: \[x = \frac{-15}{-3}\]
• Simplify the right side to get: \[x = 5\]

This simple division allows us to determine that the value of \(x\) is 5. The process of solving equations often involves a combination of operations such as addition, subtraction, multiplication, or division. The aim is to isolate and solve for the variable step by step. Practice will help you get faster and more accurate in solving equations.

Verifying Solutions

After calculating the value of the variable, it's crucial to verify the solution. Verifying ensures that the value of \(x\) satisfies the original equation. Substituting the solution back into the original equation \[-3x + 8 = -7\], we check if the left side equals the right side when \(x = 5\).

• Substitute \(x = 5\) into the equation: \[-3(5) + 8 = -7\]
• Simplify: \[-15 + 8 = -7\]
• Check equivalence: \[-7 = -7\]

Since both sides match, our solution \(x = 5\) is verified as correct. Verifying solutions helps prevent errors and confirms the integrity of your solution, allowing you to be confident with your answer.