Problem 132 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether 2 is a solution of \(13 x+3=3(5 x-1)\) Simplify: \(5(3 x+2 y)+6(5 y)\). (Section 1.4, Example 11)

Step-by-Step Solution

Answer

The value of 2 is not a solution to the first equation. The simplified form of the given expression is \(15x + 40y\).

Step 1: Substitute x=2 in the equation

Replace \(x\) with 2 in the equation \(13 x+3=3(5 x-1)\), which becomes \(13*2 + 3 = 3*(5*2 - 1)\). After simplification, this will turn into 29 = 27, which doesn't hold true. Hence, 2 is not a solution to the equation.

Step 2: Apply the Distributive Property

In the expression \(5(3 x + 2 y) + 6(5 y)\), apply the distributive property to multiply each term inside the brackets by the number outside the brackets. This results in \(15x + 10y + 30y\).

Step 3: Combine like terms

In the resulting expression, combine the terms that have the same variable and exponent, which are the \(y\)-terms in this case. This gives the final simplified expression as \(15x + 40y\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle of algebra, which allows us to multiply a single term by each term within parentheses. It's especially useful in simplifying expressions and solving equations.
For instance, in our original problem, we had the expression \(5(3x + 2y) + 6(5y)\). Using the distributive property, you multiply 5 by both \(3x\) and \(2y\), yielding \(15x + 10y\). Similarly, you do the same with the second part by multiplying 6 by \(5y\), leading to \(30y\).

• Here, 5 was distributed over \(3x\) and \(2y\)
• 6 was distributed over \(5y\)

This process transforms the original complex expression into a more manageable form without the parentheses, facilitating further simplification.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In our exercise, we needed to determine if \(x = 2\) was a solution to the equation \(13x + 3 = 3(5x - 1)\).
First, you substitute 2 for \(x\) in the equation, resulting in the calculation \(13 \times 2 + 3 = 3(5 \times 2 - 1)\). When you resolve this, the left side becomes 29 and the right side becomes 27. Since 29 does not equal 27, \(x = 2\) is not a solution.

• Substitute the variable with the proposed number.
• Calculate both sides of the equation to verify if they are equal.

If they are not equal, the number is not a solution, just like in our case where \(x = 2\) did not satisfy the equation.

Combining Like Terms

Combining like terms is a key step in simplifying algebraic expressions. It involves merging terms that have identical variable parts and powers.
For example, once you apply the distributive property to \(5(3x + 2y) + 6(5y)\), you end up with \(15x + 10y + 30y\). Here, \(10y\) and \(30y\) are like terms because they both contain the variable \(y\) raised to the same power.

• Identify terms that have the same variable.
• Add or subtract the coefficients of these like terms.

Consequently, \(10y + 30y\) simplifies to \(40y\), yielding the final expression \(15x + 40y\). This process makes the equation simpler and clearer, showcasing all terms in a neat and consolidated manner.