Problem 25 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ -5(3-2 x)-(1-x)=4(x-3) $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{4}{7} \).

Step 1: Distribute Negative Signs

Begin by distributing the negative signs in the equation. For the left side, distribute \(-5\) across \((3-2x)\) and \(-(1-x)\). This gives us:\[-5 \times 3 + 5 \times 2x - (1) + (x)\]Simplifying the expression results in:\[-15 + 10x - 1 + x = 4(x-3)\]

Step 2: Simplify and Combine Like Terms

Combine all like terms on the left side of the equation.\[-15 - 1 + 10x + x \]Simplify to:\[-16 + 11x\]The equation now is:\[-16 + 11x = 4(x-3).\]

Step 3: Expand the Right Side

Expand the right-hand side by distributing 4 across \((x-3)\):\[4 \times x - 4 \times 3 = 4x - 12\]The equation now becomes:\[-16 + 11x = 4x - 12.\]

Step 4: Move Variables to One Side

Subtract \(4x\) from both sides to get all terms involving \(x\) on one side of the equation.\[11x - 4x = -12 + 16\]This simplifies to:\[7x = 4\]

Step 5: Solve for x

To isolate \(x\), divide both sides by 7:\[x = \frac{4}{7}\]

Step 6: Check the Solution

Substitute \(x = \frac{4}{7}\) back into the original equation to verify it satisfies the equation. Left side: \(-5(3 - 2 \cdot \frac{4}{7}) - (1 - \frac{4}{7})\)Right side: \(4(\frac{4}{7} - 3)\)Calculate both sides and verify they are equal:- Left side simplifies to:\(-5 \times \frac{11}{7} - \frac{3}{7} = -\frac{61}{7}\)- Right side simplifies to:\(4 \times -\frac{17}{7} = -\frac{61}{7}\)Both sides are equal, thus verifying the solution is correct.

Key Concepts

Distributive Property

The distributive property is a key concept when solving linear equations, especially when you encounter parentheses in an equation. This property allows you to multiply a single term by each term within a set of parentheses. It is expressed as:

• For a term \(a\) outside the parentheses and terms \(b\) and \(c\) inside, the expression \(a(b + c)\) becomes \(ab + ac\).

In the original exercise equation, \(-5\) was distributed across the terms \(3 - 2x\), which resulted in:

• \(-5 \times 3 = -15\)
• \(-5 \times (-2x) = 10x\)

This careful distribution helps break down the expression into simpler parts you can easily manage. Likewise, distributing across \(1-x\) involved realizing that every term is affected by the sign on the outside, ensuring every part of the expression is accounted for. Breaking down complex equations using the distributive property makes them easier to handle, leading to better understanding and accurate solutions.

Combining Like Terms

Once you've used the distributive property, the next step is to simplify the equation by combining like terms. Like terms are terms that contain the same variable raised to the same power. In simpler terms:

• Combine all the constant numbers, those without variables.
• Combine all the terms with the same variable.

In our equation, combining like terms meant:

• Adding \(-15\) and \(-1\) to get \(-16\)
• Adding \(10x\) and \(x\) to get \(11x\)

It's crucial to ensure the signs before each term are correctly accounted for. For instance, subtracting 1 is just like adding a negative 1. After combining, you achieve a more simplified equation that makes it easier to isolate the variable and find a solution. This step is essential for reducing complexity and ensuring you are only working with the necessary components to find an answer.

Verifying Solutions

After finding a solution to an equation, it is important to verify that this solution is correct. Verification is the process of substituting the solution back into the original equation to ensure both sides equal each other.

• Substitute the found value back into the original equation.
• Calculate both the left and right side of the equation separately.
• Check that both sides are the same to confirm the solution is correct.

For the solution \(x = \frac{4}{7}\), substituting it back showed:

• The left side simplification to \(-\frac{61}{7}\)
• The right side simplification also to \(-\frac{61}{7}\)

When both sides equate, it confirms the solution is valid. This step should not be skipped as it assures accuracy of the solution.