Equation of a Straight Line with Gradient 1 Passing Through (2,3)

In this article, we will explore the different forms of the equation of a straight line with a gradient (slope) of 1 passing through the point (2,3). Whether you're working on geometry, graphing linear equations, or learning basic algebra, understanding these concepts will prove invaluable. We will cover the slope-intercept form, point-slope form, and general and standard forms of the line's equation.

Slope-Intercept Form: y mx b

The slope-intercept form of a line's equation is y mx b, where:

m represents the slope or gradient of the line, which is given to be 1 in this case. b represents the y-intercept, which is the value of y at the point where x 0.

Given that the point (2,3) lies on the line, we can substitute these values into the slope-intercept form to solve for b:

3 1(2) b

Simplifying this, we get:

b 3 - 2 1

Therefore, the equation of the line in slope-intercept form is:

y x 1

Point-Slope Form: y - y1 m(x - x1)

The point-slope form of a line's equation is y - y1 m(x - x1), where:

y1 and x1 are the coordinates of the point the line passes through. m is the slope or gradient of the line.

Using the given point (2,3) and the slope 1:

y - 3 1(x - 2)

General Form: Ax By C 0

The general form of a line's equation is Ax By C 0. We can convert the slope-intercept form to the general form by rearranging the equation:

y x 1

Multiplying both sides by 1 to clear the fraction and moving everything to one side:

1y - 1x - 1 0

Which simplifies to:

-x y - 1 0

Multiplying the whole equation by -1 to make the coefficient of x positive:

x - y 1 0

Therefore, in standard form, the equation is:

x - y 1 0

Standard Form: Ax By C

The standard form of the equation of a line is Ax By C. From the general form, we have:

x - y 1 0

Moving the constant term to the other side to match the standard form:

x - y -1

x - y -1

Conclusion

In summary, the equation of a straight line with a gradient of 1 passing through the point (2,3) can be represented in the following forms:

Slope-Intercept Form: y x 1 Point-Slope Form: y - 3 x - 2 Standard Form: x - y -1

Understanding the different forms of a line's equation is crucial for not only solving mathematical problems but also for visualizing and interpreting the graphical representation of linear relationships.