Problem 17 - Fundamentals of Mathematics

Question

Verify that each given value is a solution to the given equation. $$-6 a+3+3 a=4 a+7-3 a, a=-1$$

Step-by-Step Solution

Answer

Yes, \( a = -1 \) is a solution to the equation.

Step 1: Substitute the Given Value into the Equation

Given the equation \[-6a + 3 + 3a = 4a + 7 - 3a\]we'll substitute the value \( a = -1 \) into the equation. This gives us:\[-6(-1) + 3 + 3(-1) = 4(-1) + 7 - 3(-1)\]

Step 2: Simplify Both Sides of the Equation

Simplify each side of the equation with the substituted value:Left side: \[-6(-1) + 3 + 3(-1) = 6 + 3 - 3 = 6\]Right side:\[4(-1) + 7 - 3(-1) = -4 + 7 + 3 = 6\]

Step 3: Compare Both Sides of the Equation

Now that both sides of the equation have been simplified, compare them:Left side = 6Right side = 6Since both sides are equal, the original value \( a = -1 \) satisfies the equation.

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving equations. It's like trying a potential solution to see if it works. Here's how it works: take an equation and replace the variable with a specific value. In this exercise, we have the equation \(-6a + 3 + 3a = 4a + 7 - 3a\) and we're testing if \(a = -1\) is a valid solution.

• Identify the equation and the value to check.
• Insert, or 'substitute', the value into the equation.
• Write the equation using the substituted value.

This gives us \(-6(-1) + 3 + 3(-1) = 4(-1) + 7 - 3(-1)\). Each term with \(a\) is replaced with \(-1\), converting it from a general equation into something specific we can work with.
Next, it's time to simplify!

Simplifying Expressions

Simplifying expressions is all about making things easier to handle. It's like tidying up a room – everything becomes clearer! Once we've substituted a value into our equation, we must simplify both sides to see if they truly balance. Let's go over how to simplify.

• Look for terms you can combine on each side of the equation.
• Perform any arithmetic needed - multiply, add, or subtract.
• Ensure the equation is as simple as possible, usually combining all like terms.

In our example, the left side gives \(-6(-1) + 3 + 3(-1) = 6 + 3 - 3 = 6\), and on the right side, we get \(4(-1) + 7 - 3(-1) = -4 + 7 + 3 = 6\). Each calculation helps verify the correctness, yielding both sides to be \(6\). This signifies the equation is balanced.

Verification of Solutions

Verification of solutions is the final check in solving an equation. It's about confirming the steps and ensuring everything is correct! After substituting and simplifying, the last task is to make sure that both sides of the equation match.

• Observe the simplified results from both sides of the equation.
• If they are equal, the substituted value is a valid solution.
• If not, it means the value does not satisfy the equation.

For our equation, comparing each side reveals both equal \(6\), thus verifying \(a = -1\) as a solution. This shows that mathematically, our steps and substitutions were accurate! It's crucial to ensure a symmetrical balance, affirming our math is sound and correct.