Problem 8 - Thinking Mathematically

Question

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(3 x=2 y+6\)

Step-by-Step Solution

Answer

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

Step 1: Find the \(y\)-intercept

To find the \(y\)-intercept, set \(x=0\) in the equation. So, start with the original equation \(3 x=2 y+6\), and substitute \(0\) for \(x\). Simplify to find the value of \(y\) which is \(y=-3\). Thus, the \(y\)-intercept is \(-3\). This is the point where the line crosses the \(y\)-axis.

Step 2: Find the \(x\)-intercept

To find the \(x\)-intercept, set \(y=0\) in the equation. Start with the original equation \(3 x=2 y+6\), substitute \(0\) for \(y\) and simplify to find the value of \(x\) which is \(x=2\). Thus, the \(x\)-intercept is \(2\). This is the point where the line crosses the \(x\)-axis.

Step 3: Plot the intercepts and the line

Plot the \(y\)-intercept at \((0, -3)\) on the graph and the \(x\)-intercept at \((2, 0)\). Now, draw a straight line that passes through these two points. This line is the graph of the equation \(3 x=2 y+6\).

Key Concepts

Understanding X-Intercepts

When we talk about the x-intercept, we are focusing on the point where a line crosses the x-axis in a graph. At this particular point, the value of y is always zero.
This means to find the x-intercept of a linear equation, you need to set y to zero in the equation and solve for x.
For the equation given in the exercise, which is \(3x = 2y + 6\), setting y to zero simplifies it to \(3x = 6\).
By solving this equation, you'll find that the x-intercept is at \(x = 2\).

• Remember, x-intercept always happens at \((x, 0)\) on the graph.
• It represents the value of x when y is zero.

Understanding Y-Intercepts

The y-intercept is a key concept just like the x-intercept, but it represents where the line crosses the y-axis. At this special point, the value of x is always zero.
So, to find the y-intercept of a linear equation, you need to set x to zero in the equation and solve for y.
In the case of the equation \(3x = 2y + 6\) from the exercise, setting x to zero changes the equation to \(-2y = 6\). This simplifies the problem, showing that \(y = -3\).
Thus, the y-intercept for this equation is at \((0, -3)\).

• Y-intercept will always appear at \((0, y)\) on the graph.
• It shows the value of y when x is zero.

Solving Linear Equations for Graphing

To graph linear equations effectively, it's very useful to start with finding both the x-intercept and y-intercept. They provide two specific points through which you can draw the line of the equation.
By doing this, you turn an abstract equation, like \(3x = 2y + 6\), into something visual and comprehensible.

• First, you will determine the y-intercept by setting x to zero and solve for y.
• Second, you will find the x-intercept by setting y to zero and solve for x.
• With these two intercepts plotted on the graph, you can connect them with a straight path, revealing the line of the equation.

This method highlights the importance of intercepts in understanding linear equations thoroughly. They give clear and easily identifiable points that help in sketching out the behavior of linear relationships visually.
Having a firm grasp on intercepts and their use in graphing equips you with a solid foundation for working with more complex functions later on.