Problem 34 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(-4(x-9)=-8(x+3)\)

Step-by-Step Solution

Answer

The solution is x = -15. The equation is consistent.

Step 1: Distribute the constants

First, distribute -4 throughout (x-9) and -8 throughout (x+3). This gives -4x + 36 = -8x - 24.

Step 2: Move variables to one side

Add 8x to both sides of the equation to get the variable term on one side. This changes the equation to 4x + 36 = -24.

Step 3: Isolate the variable term

Subtract 36 from both sides to isolate the variable term. This gives 4x = -60.

Step 4: Solve for the variable

Divide both sides by 4 to solve for x. So, x = -15.

Step 5: Check the solution

Substitute x = -15 back into the original equation to verify the solution. The left side becomes -4(-15-9) = -4(-24) = 96 and the right side becomes -8(-15+3) = -8(-12) = 96. Since both sides are equal, the solution is correct.

Step 6: Conclusion on the nature of equation

Because the solution holds true, this equation is a consistent equation, but not an identity or a contradiction.

Key Concepts

Distributive Property

The distributive property allows you to simplify expressions by distributing a multiplier across terms inside parentheses. For the equation \(-4(x-9)=-8(x+3)\), start by multiplying -4 with both x and -9, and -8 with both x and 3. This gives: \(-4x + 36 = -8x - 24\). Breaking down each step helps:

• Multiply -4 by x: \(-4x\)
• Multiply -4 by -9: \(+36\)
• Multiply -8 by x: \(-8x\)
• Multiply -8 by 3: \(-24\)

After applying these multiplications, our simplified equation is \(-4x + 36 = -8x - 24\). This illustrates how the distributive property makes equations easier to work with.

Isolating the Variable

The next step is to isolate the variable, which means getting x by itself on one side of the equation. To accomplish this, you must first move all x terms to one side.

Add 8x to both sides:
\(-4x + 8x + 36 = -24\). It simplifies to \(4x + 36 = -24\). Now you have all x terms on one side. The next step is to isolate x further by eliminating any constant on the same side.

Subtract 36 from both sides:
\(4x + 36 - 36 = -24 - 36\), which simplifies to \(4x = -60\). The variable term (4x) is now isolated, and you can solve for x by dividing both sides by 4:
\(x = -15\).

Checking the Solution

Verification is crucial to ensure the solution is correct. Substitute x = -15 back into the original equation to check if both sides are equal:
For the left side: \(-4(-15-9)\) simplifies to \(-4(-24) = 96\).
For the right side: \(-8(-15+3)\) simplifies to \(-8(-12) = 96\).

Since \(96 = 96\), the original equation holds true when x is -15. Always substitute your solution back into the original equation to verify accuracy. This step ensures no errors were made during calculations.

Identities vs. Contradictions

An equation can be categorized as an identity, a contradiction, or consistent:

• An Identity is true for all values of the variable. Example: \(x + x = 2x\).
• A Contradiction is never true regardless of the variable's value. Example: \(x + 1 = x - 1\).
• A Consistent equation has specific solutions, just like our example, x = -15.

Since our solution holds true for x = -15 and doesn't fit the entire variable range, it’s considered a consistent equation that is neither an identity nor a contradiction.