Introduction to Linear Equations

Linear equations form a foundational part of mathematics and are widely used in many fields such as physics, engineering, and economics. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable raised to the first power. The equation for today’s discussion is: 8y 3x - 11.

What Does 8y 3x - 11 Mean?

This equation describes a linear relationship between two variables, x and y, with two constants: 3 and -11. The coefficient 3 is the slope of the line, indicating how much y changes for a unit change in x. The constant term -11 is the y-intercept, which is the point where the line crosses the y-axis.

How to Solve for x?

To find the value of x in terms of y, we start by rearranging the given equation: 8y 3x - 11.

Step 1: Adding 11 to Both Sides

We need to isolate the term containing x on one side of the equation. Therefore, we add 11 to both sides of the equation to balance it:

8y 11 3x - 11 11

8y 11 3x

Step 2: Dividing Both Sides by 3

Now that we have 3x on one side, we divide both sides by 3 to isolate x:

(8y 11) / 3 3x / 3

x (8y 11) / 3

This is the solution in terms of x. By substituting any value of y, you can find the corresponding value of x.

Understanding the Result

The Variable x

The equation x (8y 11) / 3 tells us that the value of x depends on the value of y. This is a linear relationship, where for every unit increase in y, x increases by (8/3) units. Additionally, when y is 0, x becomes (11/3), which is the x-intercept.

The Importance of Slope and Intercept

The slope of the line represented by this equation is (8/3), indicating the rate of change. The y-intercept is -11/3, which represents the value of y when x equals 0.

Practical Applications

Linear relationships like this have numerous real-world applications:

Physics: In kinematics, the relationship between distance, time, and velocity can often be represented by linear equations. Economics: Linear equations are used to model supply and demand curves. Engineering: In electrical circuits, Ohm's law is a direct application of linear equations to relate voltage, current, and resistance.

Conclusion

Solving for x in the equation 8y 3x - 11 is a fundamental skill in algebra. By rearranging the equation and understanding the principles of slope and intercept, we can interpret and apply this relationship in various contexts.

Further Reading

To deepen your understanding, consider exploring the following topics:

Linear Functions and Graphs Slope-Intercept Form Applications of Linear Equations in Real Life

By mastering linear equations, you pave the way for more advanced mathematical concepts and practical problem-solving skills.