A linear equation with two unknowns is an equation that can be written in the standard form A x plus B y equals C, where x and y are the variables or unknowns, and A, B, and C are constants. The graph of such an equation is always a straight line in a two-dimensional coordinate system. For example, the equation 2x plus 3y equals 6 represents a straight line.
Let's examine the components of a linear equation with two unknowns. In the standard form A x plus B y equals C, x and y are the variables or unknowns that we want to find. A, B, and C are constants - fixed numbers that define the specific equation. An important condition is that A and B cannot both be zero. If they were both zero, the equation would become zero equals C, which is either always false or meaningless.
The graph of a linear equation with two unknowns is always a straight line in a two-dimensional coordinate system. Different values of the constants A, B, and C create different lines with varying slopes and positions. The slope of the line depends on the ratio of coefficients A and B, while the constant C determines where the line intersects the axes. Each line represents all the possible solutions to that particular equation.
Let's examine a specific example: 2x plus 3y equals 6. In this equation, A equals 2, B equals 3, and C equals 6. To understand the line, we can find several points that satisfy the equation. When x equals 0, we get 3y equals 6, so y equals 2, giving us the point (0, 2). When y equals 0, we get 2x equals 6, so x equals 3, giving us the point (3, 0). We can also find that when x equals 1.5, y equals 1. All these points lie on the same straight line.
To summarize, a linear equation with two unknowns follows the standard form A x plus B y equals C, where x and y are variables, A, B, and C are constants, and A and B cannot both be zero. The graph is always a straight line. These equations have many real-world applications in economics for modeling supply and demand relationships, in physics for describing motion, and in engineering for representing design constraints. Understanding linear equations with two unknowns is fundamental for solving more complex mathematical problems.