A linear function of two variables is a mathematical expression containing two variables, typically x and y, where each variable has a maximum degree of one. The general form is y equals k x plus b, where k and b are constants. The graph of such a function is always a straight line.
Here are some examples of linear functions. Y equals 2x plus 3 has a steep positive slope. Y equals negative x plus 1 has a negative slope going downward. Y equals 0.5x minus 2 has a gentle positive slope. All linear functions share common properties: they always form straight lines, have constant rates of change, and their graphs are determined by the slope k and y-intercept b.
The slope k represents the steepness of the line, calculated as rise over run. It shows how much y changes for each unit change in x. The y-intercept b is where the line crosses the y-axis when x equals zero. A positive slope means the line goes upward, while a negative slope means it goes downward. The larger the absolute value of k, the steeper the line becomes.
Linear functions have many practical applications in everyday life. They can model distance versus time when traveling at constant speed, cost versus quantity in fixed-rate pricing, and temperature conversions like Celsius to Fahrenheit. These relationships help us predict and calculate values in real situations, making linear functions one of the most useful mathematical tools.
To summarize, a linear function of two variables has the form y equals k x plus b, where x and y are variables with maximum degree of one. The graph is always a straight line, where k represents the slope and b represents the y-intercept. Linear functions are fundamental mathematical tools with wide applications in science, economics, and everyday problem solving, making them essential to understand.