Problem 25 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 2. $$ 1.6 a+(-4)=0.032 $$

Step-by-Step Solution

Answer

The solution to the equation is \( a = 2.52 \).

Step 1: Isolate the variable

Begin by isolating the variable \( a \) by eliminating the constant term from the left side. To do this, add 4 to both sides of the equation:\[ 1.6a + (-4) + 4 = 0.032 + 4 \]\[ 1.6a = 4.032 \]

Step 2: Solve for the variable

Now that the equation is simplified to \( 1.6a = 4.032 \), solve for \( a \) by dividing both sides by 1.6:\[ a = \frac{4.032}{1.6} \]When you divide, you get:\[ a = 2.52 \]

Step 3: Check the solution

Substitute \( a = 2.52 \) back into the original equation to ensure it satisfies the equation:Original equation: \( 1.6a + (-4) = 0.032 \)Substitute \( a = 2.52 \):\[ 1.6(2.52) - 4 = 0.032 \]Calculate \( 1.6 \times 2.52 = 4.032 \) and then:\[ 4.032 - 4 = 0.032 \]Since both sides of the equation are equal, \( a = 2.52 \) is the correct solution.

Key Concepts

Isolating the Variable

When solving linear equations, one of the primary goals is to isolate the variable. This means you want to get the variable by itself on one side of the equation, usually the left side, to determine its value. In our exercise, the equation is given as \(1.6a + (-4) = 0.032\). Here, the variable is \(a\), and it is being multiplied by 1.6 and has a -4 added to it.

• First, we need to get rid of the -4 that is on the left side. We do this by adding 4 to both sides of the equation. This removes the -4 from the left side since -4 plus 4 equals zero.
• The equation then simplifies to \(1.6a = 4.032\).
• Now, the variable \(a\) is almost isolated; it is still multiplied by 1.6. To fully isolate \(a\), divide both sides of the equation by 1.6.

This approach ensures the variable stands alone and you can understand what value it must take to satisfy the equation.

Checking the Solution

After finding a solution for a variable, it is crucial to check your work and verify that the solution is correct. This means you substitute the computed value back into the original equation to ensure that both sides of the equation remain equal.

• Start with the original form of the equation: \(1.6a + (-4) = 0.032\).
• Insert the solution \(a = 2.52\) into the equation and perform the arithmetic.
• Calculate \(1.6 \times 2.52\) which gives \(4.032\), and then subtract 4 to get \(0.032\).

By performing this check, you confirm that your solution is correct and valid, reinforcing your understanding and ensuring no errors were made in the calculation process.

Algebraic Manipulation

Algebraic manipulation is a set of techniques used to rearrange and simplify equations to make them easier to solve. They include adding, subtracting, multiplying, and dividing terms consistently across an equation.

• Addition or subtraction is used to remove or balance terms on either side of the equation, helping in isolating the variable.
• Multiplication or division is employed to adjust coefficients or remove them altogether, making the variable easily solvable.
• Remember to perform each operation equally on both sides of the equation to maintain balance.

In our example, we used these manipulations by adding to remove the constant (-4) from one side and dividing to handle the coefficient (1.6) of \(a\). By mastering these algebraic techniques, you'll be well-equipped to tackle a variety of linear equations with confidence.