91Ó°ÊÓ

When solving linear equations, one of the first and most crucial steps is the isolation of variables. This involves rearranging the equation so that the variable we want to solve for stands alone on one side of the equation. Let's break this down with an example.

Consider the equation \( -15 = -52 + 1.6x \). Here, we need to isolate the variable \( x \). To start, we aim to get rid of the constant term, \( -52 \), on the side with the variable. We do this by performing the opposite arithmetic operation. Adding 52 to both sides changes the equation to \( -15 + 52 = 1.6x \). This simplifies to \( 37 = 1.6x \).

In the equation \( 7 - 3x = 52 \), we will isolate \( x \) by first subtracting 7 from both sides of the equation. This simplifies to \( -3x = 45 \).

In these examples, removing the constant terms is key. It sets the stage for you to deal directly with the variable you're solving for.

Once you've found potential solutions by isolating and solving for the variable, it's essential to verify these solutions. Verification involves substituting the value back into the original equation to ensure it holds true.

For instance, after isolating and solving for \( x \) in our first example, we found \( x = 23.125 \). To verify, substitute \( x \) back into the original equation, \( -15 = -52 + 1.6x \). After calculating \( 1.6 \times 23.125 = 37 \) and substituting, you'll find that \( -52 + 37 = -15 \). Since both sides of the equation are equal, the solution is verified.

Similarly, for the equation \( 7 - 3x = 52 \), substituting \( x = -15 \) yields \( 7 - 3(-15) = 52 \). This becomes \( 7 + 45 = 52 \), confirming that \( 52 = 52 \). Both verifications show that the solutions are correct.

Verification is a critical step, as it ensures accuracy and confidence in your computed solutions.

Arithmetic operations are fundamental in manipulating equations. The key arithmetic operations include addition, subtraction, multiplication, and division. Each plays a critical role in solving linear equations.

During equation solving, we often use arithmetic operations to eliminate constants or coefficients attached to variables. This is evident in how we isolated variables in earlier examples. For the equation \( 37 = 1.6x \), division is used to isolate \( x \). Divide both sides by 1.6, resulting in \( x = \frac{37}{1.6} \).

In another scenario, subtracting 7 from both sides in the equation \( 7 - 3x = 52 \) was necessary to bring the constant to the other side, simplifying it to \( -3x = 45 \). Then, we use division again to solve for \( x \).

By mastering these basic arithmetic operations, you can efficiently manipulate and solve equations, making them a vital part of any math toolkit.