Problem 51 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$6 x+14=2 x-2$$

Step-by-Step Solution

Answer

The solution to the equation \(6x + 14 = 2x -2\) is \(x = -4\).

Step 1: Isolate the variable

Subtract \(2x\) from both sides of the equation to get \(4x + 14 = -2\). This is done to have all terms containing \(x\) on one side of the equation.

Step 2: Continue to isolate the variable

Subtract 14 from both sides of the equation to get \(4x = -16\). This step pulls the constant term away from the variable term on the left side.

Step 3: Solve for x

Divide both sides of the equation by 4 to get \(x = -4\). This step gets \(x\) by itself to find its final value.

Step 4: Check the solution

Substitute \(x = -4\) back into the original equation \(6x + 14 = 2x - 2\). This gives \(6(-4) + 14 = 2(-4) - 2\) which simplifies to \(-24 + 14 = -8 - 2\) and then to \(-10 = -10\), confirming that the solution is correct.

Key Concepts

Addition Property of Equality

The addition property of equality is a simple yet powerful concept used when solving linear equations. This property states that if we add or subtract the same number from both sides of an equation, the equation remains balanced. Think of it like a seesaw: if you place equal weights on both sides, balance is maintained.
In the given problem,

• we started with the equation \(6x + 14 = 2x - 2\). The first step was to subtract \(2x\) from both sides. This action applied the addition property of equality.
• The goal was to move all \(x\) terms to one side to eventually isolate the variable. By subtracting \(2x\) from both sides, the equation became \(4x + 14 = -2\).
• Next, the constant \(14\) was subtracted from both sides of the equation, resulting in \(4x = -16\). Again, this step used the addition property to maintain equilibrium.

Utilizing this property helps systematically reduce and simplify equations to more workable forms, paving the way for further solving steps.

Multiplication Property of Equality

Once we have isolated a term with the variable, the multiplication property of equality helps get to the variable's pure form. This property ensures that when we multiply or divide both sides of an equation by the same non-zero number, the equality still holds.
In our exercise,

• after subtracting the necessary terms using the addition property, we arrived at \(4x = -16\). The next step was to eliminate the coefficient attached to \(x\).
• Dividing both sides by 4 was the key action, hence applying the multiplication property. This leaves \(x\) by itself on one side, resulting in \(x = -4\).

This step is crucial as it transforms an expression into a solution. Remember, always divide by the exact coefficient when using the multiplication property, ensuring the purity of the variable value.

Checking Solutions

Finally, checking solutions confirms the accuracy of your calculated value. It's a step you should never skip as it firms up your solution's reliability.
The verification process involves substituting your solution back into the original equation. In the case of our example,

• we calculated \(x = -4\).
• Substituting \(x = -4\) into the original equation \(6x + 14 = 2x - 2\) gives us \(6(-4) + 14 = 2(-4) - 2\).
• Solving both sides, we find \(-24 + 14 = -8 - 2\), which simplifies to \(-10 = -10\).

The sides of the equation match, confirming that our solution is indeed correct. This step not only reassures correct calculations but also reinforces understanding of the relationships within the equation.