Problem 76 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$5(1.2 x+6)=7.1 x+34.4$$

Step-by-Step Solution

Answer

The solution to the given equation is x = -4.

Step 1 - Distribute on the Left-Hand Side

Multiply 5 with each term inside the parentheses on the left-hand side of the equation: \(5*(1.2x) + 5*6\) which gives \(6x + 30\).

Step 2 - Group Like Terms

Put the x terms and constant terms on their own sides: \(6x - 7.1x = 34.4 - 30\).

Step 3 - Simplify

Simplify each side of the equation: \(-1.1x = 4.4\).

Step 4 - Solve for x

Divide both sidess of the equation by -1.1 to solve for x: \(x = -4\).

Step 5 - Check Your Solution

Substitute x = -4 into the original equation to verify if both sides are equal. Left side: \(5*(1.2*(-4) + 6) = -14 \). Right side: \( 7.1*(-4) + 34.4 = -14 \). Both sides are equal meaning the solution is verified.

Key Concepts

Distributive Property

When dealing with linear equations, the distributive property is a powerful tool. It allows us to simplify expressions by distributing a number over terms inside parentheses. Consider the equation given: \(5(1.2x + 6)\). By applying the distributive property, we multiply 5 by every term within the parentheses:

• 5 multiplied by \(1.2x\) becomes \(6x\).
• 5 multiplied by 6 becomes 30.

After distributing, the equation transforms into \(6x + 30\) on the left side. This transformation helps to make the equation easier to work with. In essence, distributing allows us to remove parentheses and deal with familiar expressions that are simpler to manipulate.

Like Terms

In algebra, combining like terms is a crucial process. Like terms are terms in an equation that have identical variable parts. For example, in the equation resulting from the distribution, we have the terms \(6x\) on one side and \(7.1x\) on the other. They are like terms because they both contain the variable \(x\). To group like terms in the equation, we subtract \(7.1x\) from \(6x\), resulting in \(-1.1x\). This simplification step helps to reduce the equation into a form that is much easier to solve. Similarly, the constants on the other side (34.4 and 30) can be combined by subtracting to get 4.4. Recognizing and combining like terms is essential to solving equations efficiently.

Solution Verification

After solving an equation, it is important to verify that the solution is correct. Solution verification involves substituting the found value of \(x\) back into the original equation to check if the two sides are equal. In this exercise, we found \(x = -4\).
To verify:

• Substitute \(-4\) into the original left side: \(5(1.2(-4) + 6)\) simplifies to \(-14\).
• Substitute \(-4\) into the original right side: \(7.1(-4) + 34.4\) also simplifies to \(-14\).

Since both sides of the equation equal \(-14\), we confirm that \(x = -4\) is a correct solution. This step is crucial as it assures us that no errors were made during the calculation process.

Simplifying Equations

Simplifying equations is all about making them easier to solve by breaking them down into their simplest form. Initially, we use the distributive property and combine like terms. The task then is to reduce the equation to its simplest form:

• Once we have \(6x - 7.1x = 34.4 - 30\), we simplify to \(-1.1x = 4.4\).
• Solve for \(x\) by dividing both sides by \(-1.1\) which gives \(x = -4\).

This process of simplifying ensures that equations are manageable and solutions are reached efficiently. Simplifying not only aids in solving the equation but also helps minimize errors along the way by keeping the equation parts clear and straightforward.