Standard Form of Writing the Equation of a Straight Line: ax by c

The equation of a straight line in the form ax by c is known as the standard form. Solving it for y, we can transform it into the slope-intercept form y mx b. In this guide, we will explore how to solve for both y and x and understand the properties of the line derived from this transformation.

Transforming the Equation

To solve for y, we start with the standard form of the equation:

ax by c

Isolating y on one side of the equation:

by -ax c

y -frac{a}{b}x frac{c}{b}

This is the slope-intercept form of the equation, where:

m -frac{a}{b} (slope) b frac{c}{b} (y-intercept)

Understanding the Slope and Y-Intercept

The slope of the line is given by m -frac{a}{b}. In this context, the slope is negative because the coefficient of x is negative. This means the line will slope downwards from left to right.

The y-intercept, b frac{c}{b}, is the point where the line crosses the y-axis. For non-zero values of a and b, the line will cross the y-axis at (0, frac{c}{b}).

When c 0

If c 0, the equation simplifies to:

ax by 0

Isolating y yields:

by -ax

y -frac{a}{b}x

In this case, the line passes through the origin, and the y-intercept is zero. Therefore, the line cuts the x-axis at (frac{c}{a}, 0) and the y-axis at (0, frac{c}{b}).

Special Cases

When specific values are substituted for individual variables, different properties of the line are revealed:

When x 0:

If y c/b, the line cuts the y-axis at (0, c/b). This line is parallel to the x-axis.

When y 0:

If x c/a, the line cuts the x-axis at (c/a, 0). This line is parallel to the y-axis.

For the case where all five variables can be zero, the equation ax by c would be undefined, as dividing by zero is not allowed in mathematics.

Conclusion

The standard form of the equation of a straight line, ax by c, provides a valuable tool for understanding the slope and intercepts of a line. By transforming this equation into the slope-intercept form, we can easily derive the slope and y-intercept, giving us a clear picture of the line's behavior.

Key Points

The slope of the line is given by m -frac{a}{b}. The y-intercept of the line is given by b frac{c}{b}. If c 0, the line passes through the origin. Substituting specific values for x or y can reveal special properties of the line.

Recommended Reading

Slope-Intercept Form of a Line Parallel and Perpendicular Lines Equations of a Line