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A linear function is often expressed in the form \(f(x) = mx + c\), which is known as the slope-intercept form. This equation is essential in representing a straight line on a graph. This form clearly shows the slope \(m\) and the y-intercept \(c\).

• Slope \(m\): Describes the steepness or incline of the line. It is the change in \(y\) for each change in \(x\).
• Y-intercept \(c\): The point where the line crosses the y-axis. This value indicates where the line starts when x is zero.

Understanding the slope-intercept form helps in quickly visualizing how a line behaves on a graph. Knowing how to extract these elements allows you to sketch any linear function by defining its direction and starting point.This form also makes it easy to calculate values for the function by simply inserting x-values to find corresponding y-values.
To summarize, the slope-intercept form \(f(x) = mx + c\) is an indispensable tool for working with linear functions, facilitating everything from graphing to solving practical problems.

In mathematics, a system of equations is a collection of two or more equations with a set of variables.This step involves determining whether there exists a common solution for all the equations in the system. In our problem, we dealt with two equations derived from the linear function's conditions:

• \(m = c\)
• \(5m + c = 4\)

To solve these, we use the substitution method by replacing one variable in terms of the other from the first equation into the second. This approach simplifies the problem to a single equation with one variable, making it easier to solve.Once one variable's value is determined, it can be substituted back into any original equation to find the other variable.
Systems of equations are fundamental in mathematics, allowing us to handle simultaneous conditions or constraints.Learning how to manipulate these systems is key to unlocking complex problem-solving capabilities.

Solving linear equations is a crucial skill in algebra, focusing on finding the unknown variable while maintaining the equality. In our exercise, once we set up the system, we proceeded to solve the equations:

• Starting with \(5m + m = 4\), we combined terms: \(6m = 4\).
• By dividing both sides by 6, we isolated \(m\): \(m = \frac{4}{6} = \frac{2}{3}\).

Solving involves performing operations that maintain the equality, such as addition, subtraction, multiplication, or division, to arrive at a solution. Once \(m\) was found, we used the condition \(m = c\) to directly set \(c = \frac{2}{3}\).
This exercise demonstrates the simplicity of linear equations, where each operation brings us closer to finding the unknown values. Mastering linear equations enhances your problem-solving efficiency and builds a foundation for tackling more complex mathematical scenarios.