Problem 9 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. Graph the equation. $$ 2 x-y=4 $$

Step-by-Step Solution

Answer

The x-intercept of the line is at (2, 0) and y-intercept is at (0, -4).

Step 1: Finding the x-intercept

To find the x-intercept, set y = 0 in the original equation and solve for x. The original equation is \(2x-y=4\), so by substituting y=0 in the equation, we get \(2x-0=4\), which simplifies to \(2x=4\). Solving for x gives \(x=2\).

Step 2: Finding the y-intercept

To find the y-intercept, set x = 0 in the original equation and solve for y. The original equation is \(2x-y=4\), so by substitifying x=0 in the equation, we get \(2*0-y=4\), which simplifies to \(-y=4\). Solving for y gives \(y=-4\).

Step 3: Graphing the equation

Begin by identifying the x-intercept (2,0) and y-intercept (0,-4) on the graph. Draw a line that passes through these two points. The line represents the equation \(2x-y=4\).

Key Concepts

Understanding the X-Intercept

To find the x-intercept of a linear equation, follow these steps: set the value of "y" to zero in the equation and solve for "x". The x-intercept is the point where the graph of the equation crosses the x-axis. This means all x-intercepts will have the form

• (x, 0)

For example, let's determine the x-intercept for the equation \(2x - y = 4\). We set \(y = 0\) and substitute it into the equation, giving us \(2x - 0 = 4\). This simplifies to \(2x = 4\). Solving for "x" by dividing both sides by 2, we find that \(x = 2\).

In summary, the x-intercept for our equation is the point (2, 0), which is where the line crosses the x-axis. Understanding how to find the x-intercept helps you grasp the graphical representation of linear equations.

Finding the Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. This special point has all coordinates in the form

• (0, y)

To determine the y-intercept, set "x" to zero in the equation and solve for "y".

In our example with the equation \(2x - y = 4\), we substitute \(x = 0\) into the equation. This gives us \(2*0 - y = 4\), which simplifies to \(-y = 4\). Solving for "y" by multiplying both sides by -1, we find that \(y = -4\). So, the y-intercept is the point (0, -4).

The y-intercept is crucial since it shows where the function crosses the vertical axis, illustrating the initial value of the function when "x" is zero.

Exploring Linear Equations

Linear equations represent a straight line on a graph. The general form of a linear equation is \(Ax + By = C\), where "A", "B", and "C" are constants. Linear equations can be identified by

• having one independent variable "x" and one dependent variable "y"
• producing a straight line graph

For the equation \(2x - y = 4\), it follows this format. Such equations are extremely useful in mathematics since they model real-life situations, including economics, physics, and engineering.

The power of linear equations lies in their simplicity and predictability. By understanding how to find intercepts, we can easily graph them. Doing so provides a visual representation that can help in identifying trends and making predictions.

Graphing the equation not only verifies the solutions for intercepts but also shows the line's slope and direction. In our example, knowing the intercepts allows us to draw the line on a coordinate plane by connecting the points (2, 0) and (0, -4), effectively visualizing the entire equation.