Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$7 x+3=6(x-1)+9$$

Step-by-Step Solution

Answer

The solution to the equation \(7x+3=6(x-1)+9\) is \(x = 0\).

Step 1: Distribute on the right-hand side

Start by distributing the \(6\) on the right-hand side of the equation to get: \(7x+3= 6x-6+9\).

Step 2: Simplify both sides

Next, simplify both sides to gather like terms. On the right side, add \(-6\) and \(9\) to get \(3\), so the equation simplifies to: \(7x + 3 = 6x + 3\).

Step 3: Use the addition property of equality

To isolate \(x\), subtract \(6x\) from both sides of the equation. The new equation is now: \(7x - 6x = 3 - 3\).

Step 4: Simplify the equation

The equation simplifies to \(x = 0.\)

Step 5: Check the solution

Check the proposed solution by replacing the \(x\) in the initial equation. Both sides should be equal if the solution is correct. Doing that, \(7*0+3 = 6*(0-1)+9\) simplifies to \(3 = 3\), hence confirming that the solution is correct.

Key Concepts

Addition Property of Equality

Understanding the addition property of equality is essential for solving linear equations. Here's how it works: if you have an equation like \(a = b\), and you add the same number, say \(c\), to both sides, the equation \(a + c = b + c\) still holds true. This property is based on the idea that equal quantities remain equal if the same amount is added to each side of the equation.

Let's apply this to our exercise. We have \(7x + 3 = 6x + 3\) after initial simplification. To isolate the variable \(x\), you can apply this property by subtracting \(6x\) from both sides of the equation, essentially 'adding' a negative. This gives us \(7x - 6x = 3 - 3\). Notice how we kept the equation balanced by performing the same operation on both sides? That's the addition property of equality in action, allowing us to move towards the solution, which is \(x = 0\).

Algebraic Equation Simplification

Simplifying algebraic equations is like cleaning up a messy room; it makes finding what you need (the solution) much easier. To simplify an equation, you combine like terms and remove any unnecessary parts. Like terms are terms that share the same variable raised to the same power. For instance, \(7x\) and \(6x\) are like terms.

Prior to applying the addition property, we first simplify the given exercise by distributing \(6\) on the right-hand side, to get \(6x - 6 + 9\), then we combine the constants \( -6\) and \(9\) to form \(3\), resulting in \(7x + 3 = 6x + 3\). This tidy equation is easier to work with and sets the stage for using the addition property to find the value of \(x\). Simplification is a crucial step—without it, finding the solution becomes much harder.

Equation Solving Steps

Solving equations is a process, much like following a recipe. To find the solution, you should follow a set of steps tailored to the problem at hand. Generally, these steps include distributing any multiplied values, combining like terms, and using the properties of equality to isolate the unknown.

In our example equation, the first step was to distribute the \(6\) on the right-hand side. After simplifying both sides, we used the addition property of equality by subtracting \(6x\) from both sides. Finally, we simplified the resulting expression to determine that \(x = 0\).

To verify our solution wasn't a fluke, we substituted \(x = 0\) back into the original equation and found that both sides equaled \(3\), confirming our solution's validity. Remember, checking your work is just as important as solving the equation—it ensures your answer is accurate!