Problem 1 - College Algebra Essentials

Question

Solve and check linear equation. \(7 x-5=72\)

Step-by-Step Solution

Answer

The solution to the equation \(7x - 5 = 72\) is \(x = 11\)

Step 1: Simplify the Equation

The original equation is \(7x - 5 = 72\). First, the equation must be simplified by isolating the term containing x. To do this, add 5 to both sides to remove -5 from the left side giving the simplified equation: \(7x = 72 + 5\) which simplifies to \(7x = 77\).

Step 2: Solve for X

To isolate x, divide both sides of the equation by 7. The completed operation gives \(x = 77/7\) which simplifies to \(x = 11\).

Step 3: Verify the solution

Now, the solution x=11 needs to be verified by putting it back into the original equation \(7x - 5 = 72\). This substitution will give \(7*11-5\) on the left side which will simplify to 72, the same value as the right side hence showing that x=11 is indeed the correct solution

Key Concepts

Solving Equations

When solving equations, our goal is to find the value of the variable that makes the equation true. Equations can represent a balance between two expressions, often separated by an equals sign ( = ). In the linear equation provided, \(7x - 5 = 72\), this balance is between the left-hand side, \(7x - 5\) and the right-hand side, 72.

To maintain this balance while solving for the variable \(x\), perform the same operations on both sides of the equation. This acts like a scale where adding, subtracting, multiplying, or dividing both sides equally will not tip the balance.

Here's a simple plan you can follow:

• Identify the operations being performed on the variable and decide the reverse operations to apply.
• Slowly remove all numbers or units surrounding the variable, step by step.
• Always check the simplifying operations on both sides to keep the equation balanced.

Solving equations can sometimes feel like undoing the equation. In our example, we add 5 to both sides and then divide by 7, reversing the operations initially applied to \(x\). This helps to figure out the exact number \(x\) needs to be to keep the equation true.

Isolation of Variables

Isolation of variables is a crucial step in solving equations. It involves "freeing" the variable by itself on one side of the equation. In our original equation \(7x - 5 = 72\), we want to isolate \(x\).

To start this process, we need to eliminate any terms that involve numbers alone (without the variable) from the same side as the variable. Here, that's the \(-5\), which we cancel by adding 5 to both sides of the equation, simplifying it to \(7x = 77\).

Next, adjust the coefficient of \(x\) – which is 7 in this instance. To isolate \(x\) completely, divide both sides of the equation by 7, resulting in \(x = 11\). Now, \(x\) stands alone, showing that we have successfully isolated the variable.

• Remember: whatever operation you apply to one side, apply the same to the other side.
• Always aim to leave the variable with a coefficient of 1.
• Work systematically to simplify things as much as possible.

This method is universally applicable to all types of equations and is a fundamental skill in algebra.

Checking Solutions

Once you've found a solution, it's important to verify it. This involves substituting the result back into the original equation to ensure that both sides remain equal. Checking solutions gives you confidence that your answer is accurate.

For our equation, after solving and finding \(x = 11\), substitute 11 back into the equation \(7x - 5 = 72\). This becomes \(7 \times 11 - 5\), simplifying to 77 - 5, which equals 72. Since both sides of the equation match, you can affirm that \(x = 11\) is indeed correct.

• Substitute carefully, ensuring each step is done accurately.
• Observe if both sides result in the same number, confirming correctness.
• Develop a habit of checking regularly to ensure no errors in your final answers.

Checking solutions not only provides reassurance about your skills but also helps spot and correct any errors made during the solving process. It's an essential part of mastering algebra.