Problem 30 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(3 x-4 y-10=0\)

Step-by-Step Solution

Answer

The x-intercept is \(\left(\frac{10}{3}, 0\right)\) and the y-intercept is \((0, -\frac{5}{2})\).

Step 1: Identify the Equation Type

The given equation is a linear equation, which is generally of the form \(ax + by + c = 0\). We'll use this format to find the intercepts.

Step 2: Solve for the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation and solve for \(x\). Substitute in the equation: \(3x - 4(0) - 10 = 0\). This simplifies to \(3x - 10 = 0\). Solving for \(x\), we get \(x = \frac{10}{3}\). Therefore, the \(x\)-intercept is \(\left(\frac{10}{3}, 0\right)\).

Step 3: Solve for the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation and solve for \(y\). Substitute in the equation: \(3(0) - 4y - 10 = 0\). This simplifies to \(-4y - 10 = 0\). Solving for \(y\), we get \(y = -\frac{10}{4} = -\frac{5}{2}\). Therefore, the \(y\)-intercept is \((0, -\frac{5}{2})\).

Step 4: Graph the Equation Using Intercepts

Now use the intercepts to graph the equation. Plot the \(x\)-intercept \(\left(\frac{10}{3}, 0\right)\) and the \(y\)-intercept \((0, -\frac{5}{2})\) on the Cartesian plane. Draw a straight line through these two points, extending in both directions.

Key Concepts

X-Intercept

The term "x-intercept" refers to the point where a graph crosses the x-axis. This is where the value of the variable y is zero. By understanding this concept, you can identify how the graph behaves on the x-axis, which is crucial for graphing linear equations and analyzing their properties.

• To find the x-intercept of an equation like \(3x - 4y - 10 = 0\), set \(y = 0\), and solve for \(x\).
• Substituting \(y = 0\) in the equation gives us \(3x - 10 = 0\).
• Solve \(3x = 10\) by dividing both sides by 3, resulting in \(x = \frac{10}{3}\).
• The x-intercept is therefore \(\left(\frac{10}{3}, 0\right)\).

Finding the x-intercept helps us know exactly where the line crosses the x-axis, which is valuable for sketching the graph of the equation on the Cartesian plane.

Y-Intercept

The y-intercept is the point where the graph of an equation crosses the y-axis. At this point, the value of \(x\) is zero. The y-intercept is an important feature of linear equations, as it provides insight into where the graph crosses the y-axis.

• To find the y-intercept of the equation \(3x - 4y - 10 = 0\), set \(x = 0\) and solve for \(y\).
• By substituting \(x = 0\) in the equation, we have \(-4y - 10 = 0\).
• Solving by adding 10 to both sides and dividing by -4 gives us \(y = -\frac{5}{2}\).
• This shows us that the y-intercept is \((0, -\frac{5}{2})\).

The y-intercept provides us with a key point on the y-axis, which we can use along with the x-intercept to accurately graph the linear equation. It essentially tells us where the graph crosses the y-axis, essential for setting the overall position of the graph on the Cartesian plane.

Graphing Equations

Graphing equations, especially linear ones, visually represents their solutions. It involves plotting points on the plane and connecting them to reveal the form of the equation. For linear equations, such as \(3x - 4y - 10 = 0\), this results in a straight line.

• Start by determining key points like x-intercepts and y-intercepts.
• In our example, we've found them to be \(\left(\frac{10}{3}, 0\right)\) and \((0, -\frac{5}{2})\).
• Plot these points on a graph.
• Draw a straight line through both points to complete the graph.

By connecting the intercepts with a straight line, we can effectively visualize how the linear equation behaves across different values. Graphing is a practical skill in mathematics, letting us see relationships and trends defined by the equation.

Cartesian Plane

A Cartesian plane, or coordinate plane, consists of two axes: the horizontal x-axis and the vertical y-axis. Together, they form a grid used to plot points defined by ordered pairs \((x, y)\). This framework supports graphing different mathematical functions and equations.

• The x-axis represents the horizontal component of points, where y values are zero.
• The y-axis represents the vertical component, where x values are zero.
• The point of intersection of the x and y axes is called the origin, \((0, 0)\).
• Each point on the plane is referenced by an x-y pair, showing its location

Using a Cartesian plane, we can graph linear equations by plotting intercepts and drawing lines through these reference points. It gives a visual context to abstract algebraic equations, making concepts much more accessible.