Problem 144 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section. Find the ordered pairs \((\quad, 0)\) and \((0, \quad)\) satisfying \(4 x-3 y-6=0\).

Step-by-Step Solution

Answer

The ordered pairs satisfying \(4x - 3y -6 =0\) are \((\frac{3}{2}, 0)\) and \((0, -2)\).

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation:\(4x - 3(0) - 6 = 0 \4x - 6 = 0 \4x = 6 \x = \frac{3}{2}\)So, the x-intercept is \((\frac{3}{2}, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation: \(4(0) - 3y - 6 = 0 \-3y - 6 = 0 \-3y = 6 \y = -\frac{6}{3} \y = -2\)So, the y-intercept is \((0, -2)\).

Key Concepts

Understanding x-intercept

When working with linear equations, the x-intercept is an important point where the line crosses the x-axis. In mathematical terms, this is where the value of y is equal to zero. To find the x-intercept of any linear equation, you simply need to substitute zero for y and solve the equation for x. In our original problem, we started with the equation:

• \(4x - 3y - 6 = 0\)

By setting y to zero, we simplified the problem to find x. This gave us:

• \(4x - 6 = 0\)

Solving this, we isolated x by adding 6 to both sides and then dividing by 4, resulting in \(x = \frac{3}{2}\). This tells us that the point \((\frac{3}{2}, 0)\) is where the line hits the x-axis. Recognizing and calculating the x-intercept is crucial as it provides a point that can help graph the line on a coordinate plane.

Explaining y-intercept

The y-intercept is where a line crosses the y-axis in a coordinate system. This occurs when x equals zero, meaning the x-value does not influence the location of the intercept on the line. To find the y-intercept in a linear equation, we replace x with zero and solve for y. Using our problem's equation:

• \(4x - 3y - 6 = 0\)

Substituting zero for x simplifies the equation:

• \(-3y - 6 = 0\)

We then solve for y. By adding 6 to both sides and dividing by -3, we find \(y = -2\). Therefore, the y-intercept is \((0, -2)\). The y-intercept is an essential concept as it provides a concrete point to begin graphing our line on the y-axis. This knowledge aids in understanding the relationship between the line's equation and its graphical representation.

Understanding ordered pairs

Ordered pairs are a way to represent points on a coordinate plane, where each pair consists of an x-coordinate and a y-coordinate. They are generally written in the form \((x, y)\). This format tells us exactly where on a grid a particular point lies. The x-value indicates the horizontal position while the y-value indicates the vertical position.
For our problem, we found two crucial ordered pairs. The x-intercept \((\frac{3}{2}, 0)\) and the y-intercept \((0, -2)\). These aren't random numbers; they tell you that:

• The point \((\frac{3}{2}, 0)\) lies on the x-axis 1.5 units to the right of the origin.
• The point \((0, -2)\) lies on the y-axis 2 units below the origin.

Understanding ordered pairs is valuable as it helps visualize the points of intersection on the plane. By knowing both the x and y intercepts, you can effectively plot the initial framework for graphing the linear equation on a coordinate grid.