problems with solutions.

What is Linear Relationship?

Linear relationship is a term used to describe a mathematical equation or function. It is often referred to as a straight line relationship. A linear relationship is a type of relationship between two variables in which one variable is a function of the other. This means that when one of the variables changes, the other variable will change in a predictable way. For instance, if one variable increases, the other variable will increase in a linear fashion as well. Linear relationships are found in many areas of mathematics, including algebra, geometry, and calculus.

Examples of Linear Relationship

A classic example of linear relationship is the equation y = mx + b, where m is the slope and b is the y-intercept. This equation is used to describe a line on the Cartesian plane. Another example of linear relationship is the equation y = ax2 + bx + c, which is used to describe a parabola. Finally, an example of linear relationship is the equation y = ax3 + bx2 + cx + d, which is used to describe a cubic function.

Practical Uses of Linear Relationship

Linear relationships are used in a variety of practical applications. For instance, they are used to design bridges and buildings. They are also used to calculate the forces and moments in mechanical systems. Finally, linear relationships are used in physics to calculate the trajectories of objects in motion.

Limitations of Linear Relationship

Linear relationships are limited in their accuracy when applied to real-world situations. This is because real-world situations are often more complex than linear equations can accurately describe. For example, the equation y = ax2 + bx + c cannot accurately describe parabolic trajectories in three-dimensional space. Therefore, when attempting to solve real-world problems, other more complex mathematical equations and models may be necessary.

Sample Problems with Solutions

Problem 1:

A line has a slope of 3 and passes through the point (2, 8). Find the equation of the line.

Solution: The equation of the line is y = 3x + 2.

Problem 2:

A parabola has a vertex at (4, -1) and passes through the point (0, 5). Find the equation of the parabola.

Solution: The equation of the parabola is y = -x2 + 8x – 5.

Problem 3:

A cubic function passes through the points (1, 2), (2, 6), and (3, 7). Find the equation of the cubic function.

Solution: The equation of the cubic function is y = -x3 + 7×2 – 12x + 6.

Conclusion

Linear relationships are used to describe many mathematical equations and functions. This includes the equation of a line, parabola, and cubic function. Linear relationships are useful in a variety of practical applications, however they are limited in accuracy when applied to real-world problems. Therefore, when attempting to solve real-world problems, other more complex mathematical equations and models may be necessary.