Question

For the line \(2 x+3 y=6,\) the \(x\) -intercept is ______ and the \(y\) -intercept is ____.

Short answer

The x-intercept is (3, 0) and the y-intercept is (0, 2).

Step-by-step solution

Understand the Equation

The given equation of the line is in standard form: y = 2x + 3y = 6.

Find the X-Intercept

To find the x-intercept, set y = 0 in the equation:\[ 2x + 3(0) = 6 \]Solve for x:\[ 2x = 6 \]\[ x = 3 \].So, the x-intercept is (3, 0).

Find the Y-Intercept

To find the y-intercept, set x = 0 in the equation:\[ 2(0) + 3y = 6 \]Solve for y:\[ 3y = 6 \]\[ y = 2 \].So, the y-intercept is (0, 2).

Key concepts

x-intercept

The x-intercept of a line in a coordinate system is the point where the line crosses the x-axis. To find the x-intercept, we set y to zero in the line's equation and solve for x. This is because at any point on the x-axis, the y-coordinate is always zero.

For example, in the equation given in the exercise, \(2x + 3y = 6\), setting \(y = 0\) gives:\begin{align*} \ 2x + 3(0) = 6 \ 2x = 6 \ x = 3 \ \end{align*}
So, the x-intercept is (3, 0). This means that the line crosses the x-axis at the point where x is 3 and y is 0. Simple steps like these can always help you identify the x-intercept easily.

y-intercept

The y-intercept is where the line crosses the y-axis. To find the y-intercept, we set x to zero in the line's equation and solve for y. This is because at any point on the y-axis, the x-coordinate is always zero.

Using the equation from our exercise, \(2x + 3y = 6\), setting \(x = 0\) gives:\begin{align*} \ 2(0) + 3y = 6 \ 3y = 6 \ y = 2 \ \end{align*}
So, the y-intercept is (0, 2). This means that the line intersects the y-axis at the point where y is 2 and x is 0. Finding the y-intercept follows a similar approach to finding the x-intercept, and practicing these steps will solidify your understanding.

standard form of a line

The standard form of a line is a way of writing the equation of a line so that it is clear and organized. It is generally written as \(Ax + By = C\), where A, B, and C are integers, and A should be a non-negative integer.

In our exercise, the equation \(2x + 3y = 6\) is already in standard form. Here:

• A = 2
• B = 3
• C = 6

The standard form is helpful because it makes it easier to find both the x-intercept and the y-intercept directly. It organizes the equation in a form where algebraic manipulation and understanding intercepts become straightforward. Mastering how to work with this form will enable you to handle more complex linear equations effectively.