Problem 74 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$40+14 k=2(-4 k-13)$$

Step-by-Step Solution

Answer

The solution to the equation is \(k=-3\)

Step 1: Distribute

Start by distributing the 2 on the right side of the equation. It gives \(40+14k=-8k-26\).

Step 2: Collect like terms

Bring all \(k\) terms to one side and constants to the other side. Add \(8k\) on both sides, we get \(40+14k+8k=-26\), which simplifies to \(22k+40=-26\). Then subtract 40 from both sides which gives \(22k=-66\).

Step 3: Solve for k

To find the value of \(k\), divide both sides by 22. We get \(k=-66/22\).

Step 4: Check the solution

Now, substitute \(k=-3\) into the original equation to check if it holds true. The original equation is \(40+14*(-3)=2(-4*(-3)-13)\), which simplifies to \(40-42=-2+(-26)\). Both sides give -2, therefore the solution is correct.

Key Concepts

Distributive Property

The distributive property is a crucial tool in solving algebraic equations. It allows you to simplify expressions by multiplying a term outside a parenthesis into each term inside the parenthesis. For example, in the equation \(40 + 14k = 2(-4k - 13)\), we apply the distributive property to the right side. The term '2' is distributed to both '-4k' and '-13'. This means we multiply '2' by each term:

• The calculation becomes \(-8k - 26\).

After applying the distributive property, the equation simplifies to \(40 + 14k = -8k - 26\). This tool helps us get rid of parentheses, simplifying further solving steps.

Combining Like Terms

Combining like terms is another vital skill in solving algebra equations. It involves adding or subtracting terms that have the same variable raised to the same power. In our exercise, after using the distributive property, we have:

• Left side: \(40 + 14k\)
• Right side: \(-8k - 26\)

To simplify the equation further, move all variable terms (containing \(k\)) to one side and constants to the other. By adding \(8k\) to both sides, we combine the variable terms:

• The equation becomes \(22k + 40 = -26\).

By combining terms, especially like terms, equations become simpler and easier to solve.

Solving Linear Equations

Solving linear equations involves isolating the variable to determine its value. With our simplified equation \(22k + 40 = -26\), the next step is to bring coefficients next to the variable, and constants on the other side.

• Subtract 40 from both sides to clear the constant on the left: \(22k = -66\).

Now, divide by the coefficient of \(k\), which in this case is 22, to solve for \(k\):

• \(k = \frac{-66}{22}\), leads to \(k = -3\).

By following these steps, you can solve any linear equation where it's possible to find a unique solution.

Checking Solutions in Algebra

Once you've calculated the solution for a variable, it's essential to check your work to ensure the solution is correct. Checking the solution involves substituting the value back into the original equation to see if both sides equal. For our exercise, substitute \(k = -3\) back into the original equation:

• Left side: \(40 + 14(-3) = 40 - 42 = -2\)
• Right side: \(2(-4(-3) - 13) = 2(12 - 13) = 2(-1) = -2\)

Both sides of the equation equate to \(-2\), confirming the solution \(k = -3\) is accurate. This step ensures confidence in your result and verifies the correctness of your calculations, eliminating any doubt.