Problem 15 - Prealgebra

Question

Solve each equation. Check your solution. $$37=4 d+5$$

Step-by-Step Solution

Answer

The solution is \(d = 8\).

Step 1: Isolate the variable term

Start solving the equation by isolating the term with the variable. The equation is: \[ 37 = 4d + 5 \]Subtract 5 from both sides to remove the constant term on the right side. This gives:\[ 37 - 5 = 4d \]Which simplifies to:\[ 32 = 4d \]

Step 2: Solve for the variable

Divide both sides of the equation by 4 to solve for \(d\):\[ \frac{32}{4} = d \]This simplifies to:\[ 8 = d \]

Step 3: Verify the solution

Substitute \(d = 8\) back into the original equation to check if it satisfies the equation:\[ 37 = 4(8) + 5 \]Calculate the right-hand side:\[ 4(8) = 32 \]Then add 5:\[ 32 + 5 = 37 \]Since both sides of the equation are equal, \(d = 8\) is indeed the correct solution.

Key Concepts

Isolation of Variables

Solving algebraic equations often begins with a critical step - isolating the variable. This process involves rearranging the equation so that the variable, in this case \(d\), stands alone on one side of the equation. In the given equation \(37 = 4d + 5\), the goal is to manipulate it such that "\(d\)" is by itself.The first action is to physically "move" all other terms to the opposite side of the variable. Here's how you can do it:

• Identify the variable term: In the equation \(37 = 4d + 5\), "\(4d\)" is our variable term.
• Remove constants: Subtract 5 from both sides to eliminate the constant term from the variable side. This gives you \(37 - 5 = 4d\), simplifying to \(32 = 4d\).

By performing these operations, you've successfully isolated the variable term \(4d\), setting the stage for solving for \(d\). This not only simplifies the equation but also makes the solution path clearer at every subsequent step.

Step-by-Step Algebra

In algebra, breaking down solutions step-by-step ensures clarity and precision, making the problem more manageable. Once we've isolated the variable, the next task is to solve for the variable explicitly. Continuing with our example, you reach \(32 = 4d\). The next logical action is to resolve for \(d\):

• Division to solve: Divide both sides by the coefficient of \(d\). In this case, it’s 4, so \(32/4 = d\).
• Outcome: This operation simplifies to \(d = 8\).

Solving equations one step at a time helps you avoid mistakes and understand each operation's purpose. It’s like following a recipe where each step builds upon the previous one to bring you to a solution.

Checking Solutions

Verifying the solution of an equation is an essential step to ensure accuracy, especially in algebraic problems. After you solve the equation and find that \(d = 8\), it's crucial to check if this value truly satisfies the original problem.How do you verify?

• Substitute \(d = 8\) back into the original equation \(37 = 4d + 5\). This becomes \(37 = 4(8) + 5\).
• Simplify and compute the right side: Begin by multiplying \(4 imes 8\) to get 32.
• Add 5 to 32, resulting in 37.

When both sides of the equation equal 37, you've confirmed that the solution \(d = 8\) is indeed correct. This step not only confirms the answer but also solidifies your understanding of solving equations. It's like double-checking your work in a test to ensure no mistakes were left behind.