Question

Solve each equation (find the solution set), and check each solution. (a) \(4(6 \mathrm{x}+5)-3(\mathrm{x}-5)=0 .\) (b) \(8+3 \mathrm{x}=-4(\mathrm{x}-2)\)

Short answer

For equation (a), \(x = -\frac{5}{3}\) is the solution. We checked this by substituting it back into the original equation. For equation (b), \(x = 0\) is the solution. We also checked this by substituting it back into the original equation.

Step-by-step solution

Distribute the numbers

Multiply 4 by the terms inside the first parenthesis, and multiply -3 by the terms inside the second parenthesis: \(24x + 20 - 3x + 15 = 0\)

Combine like terms

Combine the terms with x and the constants: \(21x + 35 = 0\)

Isolate the x term

Subtract 35 from both sides of the equation: \(21x = -35\)

Solve for x

Divide both sides by 21 to find the value of x: \(x = -\frac{35}{21}\)

Simplify the solution

Simplify the fraction to obtain the solution: \(x = -\frac{5}{3}\)

Check the solution

Substitute the value of x back into the original equation to check if it is correct: \(4(6(-\frac{5}{3})+5)-3((-\frac{5}{3})-5) = 0\) This simplifies to: \(4(-3+5) - 3(-\frac{2}{3}) = 0\) Which further simplifies to: \(4(2) - 3(-\frac{2}{3}) = 0\) The expression simplifies to 0, confirming our solution is correct. (b) 8+3x=-4(x-2)

Distribute the number

Multiply -4 by the terms inside the parenthesis: \(8 + 3x = -4x + 8\)

Move the constant to one side

Subtract 8 from both sides of the equation: \(3x = -4x\)

Combine like terms

Add 4x to both sides to have all x terms on one side: \(7x = 0\)

Solve for x

Divide by 7 on both sides to find the value of x: \(x = 0\)

Check the solution

Substitute the value of x back into the original equation to check if it is correct: \(8 + 3(0) = -4(0 - 2)\) Which simplifies to: \(8 = -4(-2)\) The expression simplifies to 8=8, confirming our solution is correct.

Key concepts

Distributive Property

The distributive property is a fundamental principle in algebra and it helps in simplifying equations by eliminating parentheses. Essentially, it allows you to "distribute" multiplication across terms within parentheses. This is done by multiplying the term outside the parenthesis by each term inside. For example, in the equation \(4(6x + 5) - 3(x - 5) = 0\), apply the distributive property as follows:

• Multiply 4 by each term inside the first parentheses: \(4 \times 6x\) and \(4 \times 5\).
• Multiply -3 by each term inside the second parentheses: \(-3 \times x\) and \(-3 \times -5\).

This results in the equation: \(24x + 20 - 3x + 15 = 0\). By distributing, you have removed the parentheses, making it easier to manipulate the equation further.

Combining Like Terms

Once you've applied the distributive property, the next step is to simplify the equation by combining like terms. Like terms are terms in an equation that have the same variable to the same power.

• In our example, the terms with 'x' are \(24x\) and \(-3x\). These can be combined to yield \(21x\).
• Similarly, the constants \(20\) and \(15\) add up to \(35\).

By combining these terms, our equation simplifies to \(21x + 35 = 0\). Combining like terms makes solving the equation more straightforward by reducing the number of terms.

Isolation of Variables

To solve for the variable, we need to isolate it on one side of the equation. This helps in finding the precise value of the variable that satisfies the equation. In our exercise, we have the simplified equation \(21x + 35 = 0\). Start by moving the constant term to the other side:

• Subtract 35 from both sides: \(21x = -35\).
• Next, divide both sides by the coefficient of x, which is 21, to solve for \(x\): \(x = -\frac{35}{21}\).

This method of isolating the variable ensures that you systematically reduce the equation to find the exact value of \(x\). Simplifying the fraction, you find \(x = -\frac{5}{3}\).

Checking Solutions

The final step in solving an equation is to check your solution by substituting the found value back into the original equation. This step is crucial as it verifies the correctness of your solution.

• Substitute \(x = -\frac{5}{3}\) back into the original equation \(4(6x + 5) - 3(x - 5) = 0\).
• Calculate each part carefully: \(4(6(-\frac{5}{3}) + 5) - 3((-\frac{5}{3}) - 5)\).
• This simplifies to \(4(-10 + 5) - 3(-\frac{20}{3})\).
• After calculating, you should find the expression reduces to 0.

Double-checking like this ensures the derived solution satisfies the original equation and confirms its accuracy.