Problem 14 - Precalculus

Question

The graph of each equation is a straight line. Graph the equation after finding the \(x\)-and the \(y\) -intercepts. (since you are given that the graph is a line, you need only plot two points before drawing the line.) $$2 x-3 y=6$$

Step-by-Step Solution

Answer

The x-intercept is (3, 0) and the y-intercept is (0, -2). Plot these points and draw the line.

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation. Substitute into the equation: \(2x - 3(0) = 6\). This simplifies to \(2x = 6\). Solve for \(x\) by dividing both sides by 2, giving \(x = 3\). The x-intercept is at the point \((3, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation. Substitute into the equation: \(2(0) - 3y = 6\). This simplifies to \(-3y = 6\). Solve for \(y\) by dividing both sides by -3, giving \(y = -2\). The y-intercept is at the point \((0, -2)\).

Step 3: Plot the intercepts

Plot the points \((3, 0)\) and \((0, -2)\) on the coordinate plane. These are the points where the line will intersect the x-axis and y-axis.

Step 4: Draw the line

Using a ruler, draw a straight line through the points \((3, 0)\) and \((0, -2)\) on the graph. This line represents the equation \(2x - 3y = 6\).

Key Concepts

x-intercept

The \(x\)-intercept of a linear equation is where the graph of the line crosses the \(x\)-axis. At this intercept, the value of \(y\) is always zero, because the point lies directly on the \(x\)-axis.

To find the \(x\)-intercept, follow these steps:

• Set \(y = 0\) in the equation.
• Solve the equation for \(x\).

This gives you the point \((x, 0)\), which is the exact location on the \(x\)-axis where the graph intersects.

In our exercise, substituting \(y\) with zero in the equation \(2x - 3y = 6\) simplifies to \(2x = 6\). When solved, \(x = 3\). This means the \(x\)-intercept is at the coordinate \((3, 0)\). While this might seem straightforward, identifying the \(x\)-intercept is critical in plotting graphs and understanding their behavior.

Always remember: if a line intercepts the \(x\)-axis, the \(y\) value is zero.

y-intercept

The \(y\)-intercept is the point where a line crosses the \(y\)-axis. Here, the \(x\)-coordinate is always zero since the intersection occurs on the vertical axis.

You can find the \(y\)-intercept by following these steps:

• Set \(x = 0\) in the equation.
• Solve for \(y\).

This yields the point \((0, y)\), illustrating where the line intersects the \(y\)-axis.

In the given example of \(2x - 3y = 6\), substituting \(x\) for zero leads to \(-3y = 6\). When solved, we find \(y = -2\), providing the \(y\)-intercept at \((0, -2)\).

Recognizing the \(y\)-intercept is just as significant as finding the \(x\)-intercept. It provides a reference point that helps in sketching out the linear equation graphically, offering clues about the line's slope and its general direction.

Always remember, when a line crosses the \(y\)-axis, the \(x\) value is zero.

coordinate plane

The coordinate plane is a two-dimensional area defined by a horizontal \(x\)-axis and a vertical \(y\)-axis. This plane is a fundamental tool for graphing equations and visualizing relationships between variables.

Here's how the coordinate plane is structured:

• The point where the \(x\)-axis and \(y\)-axis intersect is known as the origin, designated as \((0,0)\).
• A point on this plane has two coordinates, \(x\) and \(y\), representing horizontal and vertical positions, respectively.
• Positive and negative values determine the direction and quadrant where the point is located.

In graphing the equation \(2x - 3y = 6\), the coordinate plane enables us to place and visualize the \(x\)- and \(y\)-intercepts. Plotting the points \((3, 0)\) and \((0, -2)\) helps in forming a straight line that represents the equation.

Understanding the coordinate plane is essential for algebra and further mathematical concepts, as it serves as a visual representation of equations. This makes complex relationships easier to comprehend and analyze.

A strong grasp on positioning points and interpreting movement along the axes is crucial in mastering graphing techniques. Make sure to get comfortable with recognizing lines, curves, and intersections as they are key aspects of exploring mathematical principles on a coordinate plane.