Problem 21 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$x-4=3 y$$

Step-by-Step Solution

Answer

The x-intercept is (4, 0) and the y-intercept is (0, -\frac{4}{3}).

Step 1: Rewrite the Equation

Rewrite the given linear equation in the form of y = mx + b. Start by isolating y. The given equation is: $$x - 4 = 3y$$Add 4 to both sides:$$x = 3y + 4$$Next, subtract 4 from both sides:$$x - 4 = 3y$$Finally, divide both sides by 3:$$y = \frac{x - 4}{3}$$

Step 2: Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Start with the equation:$$x - 4 = 3(0)$$This simplifies to:$$x - 4 = 0$$Add 4 to both sides:$$x = 4$$Therefore, the x-intercept is at (4, 0).

Step 3: Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Start with the equation:$$0 - 4 = 3y$$This simplifies to:$$-4 = 3y$$Divide both sides by 3:$$y = -\frac{4}{3}$$ Therefore, the y-intercept is at (0, -\frac{4}{3}).

Key Concepts

x-intercept

The x-intercept is where the graph of a linear equation crosses the x-axis. This point is crucial because it gives one of the key values of the equation. To find the x-intercept, set y to 0 in the equation and solve for x.

Let's look at our equation: \(x - 4 = 3y\).
When we set y to 0, the equation becomes \(x - 4 = 3(0)\).
This simplifies to \(x - 4 = 0\).
;To isolate x, add 4 to both sides: \(x = 4\).
Hence, the x-intercept of the equation \(x - 4 = 3y\) is at (4, 0).

Understanding the x-intercept helps in plotting the graph accurately and analyzing the behavior of the equation across different values of x.

y-intercept

The y-intercept is where the graph crosses the y-axis. This point is fundamental as it shows the value of y when x is 0. To find it, set x to 0 in the equation and solve for y.

For our given equation, \(x - 4 = 3y\),
we set x to 0: \(0 - 4 = 3y\).
This simplifies to \(-4 = 3y\).
Now, divide both sides by 3 to isolate y: \(y = -\frac{4}{3}\).
Thus, the y-intercept of the equation \(x - 4 = 3y\) is at (0, -\frac{4}{3}).

Knowing the y-intercept allows us to plot the starting point on the y-axis, aiding in sketching the overall graph of the linear equation.

solving linear equations

Solving linear equations often involves several steps to isolate the variable of interest. Here's a more detailed look at how to approach it using our example equation, \(x - 4 = 3y\):

• First, rewrite the equation in a more conventional form (y = mx + b), which is known as the slope-intercept form. This helps in identifying the slope and intercepts easily.
• Start by isolating y on one side of the equation. Add 4 to both sides: \(x = 3y + 4\).
• Then, subtract 4 from both sides to bring the equation into one step closer to isolating y: \(x - 4 = 3y\).
• Finally, divide every term by 3: \(y = \frac{x - 4}{3}\). Now your equation is in the y = mx + b form, where \(m\) is the slope and \(b\) is the y-intercept.

Steps like these are essential in simplifying and solving linear equations. These steps ensure that each part of the equation is correctly handled. This ensures that we get accurate intercepts and are able to graph the linear relationship effectively. Breaking down the steps helps in better understanding and application.