Problem 49 - Elementary Algebra

Question

Solve. $$ 56 x+13=32 $$

Step-by-Step Solution

Answer

x = \( \frac{19}{56} \)

Step 1: Move Constant to Other Side

First, we want to isolate the term with the variable by moving the constant on the left side of the equation to the right side. To do this, subtract 13 from both sides:\[ 56x + 13 - 13 = 32 - 13 \]This simplifies to:\[ 56x = 19 \]

Step 2: Solve for x

Now that we have isolated the term with the variable, we can solve for \( x \) by dividing both sides of the equation by 56:\[ x = \frac{19}{56} \]

Key Concepts

Solving Equations

In the realm of algebra, solving equations is a fundamental skill. It involves finding the value of a variable that makes an equation true. An equation is like a balance beam, with both sides weighing equal. Our goal is to keep this balance while finding the unknown variable, often represented by letters like \( x \), \( y \), or \( z \).

To solve an equation, we use mathematical operations like addition, subtraction, multiplication, and division. These operations help us simplify the equation and discover the value of the variable. It's crucial to apply these operations carefully to maintain the balance on both sides of the equation. As long as we perform the same operation on each side, the equality holds.

For example, consider the equation \( 56x + 13 = 32 \). Our aim is to find the value of \( x \). We'll use a series of understandable and clear steps to do this.

Isolation of Variables

Isolation of variables is an essential step when solving equations. It means arranging the equation so that the variable stands alone on one side. This makes it easy to see what the variable equals. In the expression \( 56x + 13 = 32 \), our variable, \( x \), is bundled with a constant (13) and a coefficient (56).

Start by removing the constant—here, we subtract 13 from both sides. This action removes the constant paired with the variable and helps in isolating \( x \):

• Subtract 13, so the equation becomes \( 56x = 19 \).

Now \( x \) is closer to being isolated. Next, we divide by the coefficient of 56. This further isolates \( x \), leaving us with \( x = \frac{19}{56} \).

By isolating the variable, we make the equation clearer, ensuring \( x \) is explicitly known. This is a crucial part of solving equations, allowing for simpler interpretation and solutions.

Algebraic Manipulation

Algebraic manipulation is the art of shifting terms around an equation in order to isolate the variable. It involves systematic and logical steps to simplify equations while adhering to algebraic rules.

The process begins with identifying terms that need to be moved or altered. You add, subtract, multiply, or divide these terms—always maintaining the balance of the equation.

In our example, \( 56x + 13 = 32 \), we start by moving the constant through subtraction. Algebraic manipulation helped simplify the left side:

• First, subtract 13 from both sides: \( 56x + 13 - 13 = 32 - 13 \).
• Resulting in: \( 56x = 19 \).

Next, we handle the equation's coefficient by division:

• Divide both sides by 56: \( x = \frac{19}{56} \).

Algebraic manipulation is pivotal in solving equations, making it easier to isolate the variable and find the solution.