Problem 41 - College Algebra

Question

For each of the following exercises, find and plot the \(x\) -and \(y\) -intercepts, and graph the straight line based on those two points. $$3 y=-2 x+6$$

Step-by-Step Solution

Answer

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

Step 1: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation. The equation is \(3y = -2x + 6\). Substituting \(x = 0\), we get \(3y = 6\). Solving for \(y\), divide both sides by 3 to get \(y = 2\). So, the y-intercept is \((0, 2)\).

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation. The equation is \(3y = -2x + 6\). Substituting \(y = 0\), we get \(0 = -2x + 6\). Solving for \(x\), add \(2x\) to both sides to get \(2x = 6\). Divide both sides by 2 to get \(x = 3\). So, the x-intercept is \((3, 0)\).

Step 3: Plot the intercepts

Plot the points \((0, 2)\) and \((3, 0)\) on the coordinate plane. These are the intersections with the y-axis and x-axis, respectively.

Step 4: Draw the line

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

Key Concepts

Linear Equations

In algebra, a linear equation is an equation that forms a straight line when plotted on a graph. The standard form of a linear equation is often written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. Linear equations model relationships where one variable changes at a constant rate with another.
To solve a linear equation for one variable, you often manipulate the equation to isolate the variable on one side. This allows you to express the variable in terms of known quantities. For example, you can rearrange \(3y = -2x + 6\) to solve for either \(x\) or \(y\), as needed depending on the intercepts you are trying to find.
Understanding the basics of linear equations is crucial because they form the foundation for several algebraic and calculus concepts you will learn in higher classes.

Intercepts

Intercepts are important points on a graph where the line intersects the axes. Specifically, the \(x\)-intercept is where the line crosses the x-axis, and the \(y\)-intercept is where it crosses the y-axis. These intercepts provide valuable information about the line and are useful for graphing the equation.

• Finding the \(y\)-intercept involves setting \(x = 0\) in the equation and solving for \(y\). In the example \(3y = -2x + 6\), substituting \(x = 0\) gives \(3y = 6\) which results in \(y = 2\). Therefore, the \(y\)-intercept is \((0, 2)\).
• Finding the \(x\)-intercept involves setting \(y = 0\) in the equation and solving for \(x\). Setting \(y = 0\) in \(3y = -2x + 6\) leads to \(0 = -2x + 6\), which simplifies to \(x = 3\). Hence, the \(x\)-intercept is \((3, 0)\).

Knowing how to find the intercepts is crucial for sketching the graph of a linear equation quickly.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which you can plot points, lines, and curves. It is made up of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where the axes intersect is called the origin, which is labeled as \((0, 0)\).

• Points on the coordinate plane are represented as ordered pairs \((x, y)\).
• Movement along the x-axis or y-axis represents a change in the values of variables \(x\) or \(y\), respectively.
• Understanding numbers' placement on this plane helps in identifying relationships and patterns in algebraic equations.

The coordinate plane is essential in graphing equations, including linear equations, where you can visualize the relationship between the variables as geometric shapes, mainly lines.

Graphing Lines

Graphing linear equations involves plotting points and connecting them to form a straight line. This process gives a visual representation of the equation, reflecting the relationship between the variables.
To graph lines, follow these steps:

• Identify the intercepts: We've already calculated intercepts in the form of \((0, 2)\) and \((3, 0)\).
• Plot these intercepts on the coordinate plane.
• Use a ruler to draw a line through these points, showing the path of the equation on the graph.

Visualizing equations as lines can help you understand how changes in \(x\) affect \(y\), and vice versa. This is key to mastering concepts in algebra and more complex areas such as calculus and physics.