Problem 95 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. $$3 y=6 x+3$$

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation is \(-\frac{1}{2}\), and the y-intercept is 1.

Step 1: Find the x-intercept

To find the x-intercept of the equation, substitute y = 0. into the given equation. This gives \(3(0)=6x+3 \Rightarrow 6x+3=0 \Rightarrow x=-\frac{3}{6}=-\frac{1}{2}\).

Step 2: Find the y-intercept

To find the y-intercept of the equation, substitute x = 0. into the given equation. This gives \(3y=6(0)+3 \Rightarrow 3y=3 \Rightarrow y=1.\)

Key Concepts

x-intercept

The x-intercept of a graph is the point where the graph touches or crosses the x-axis. This is the point where the value of y is zero. To find the x-intercept, you need to solve the equation for x when y is set to 0. In simpler terms, substitute 0 for y in the equation, and solve for x.
This tells you at what point the line will cut the x-axis. For the equation given:

• Let y = 0, then substitute into the equation: \(3(0)=6x+3\).
• Simplifying gives \(6x+3=0\).
• Solve for x, which results in \(x=-\frac{1}{2}\).

Therefore, the x-intercept is \(-\frac{1}{2}\),0). Think of the x-intercept as the place your line stops being vertical and starts moving horizontally across your graph.

y-intercept

The y-intercept is the point on the graph where it crosses the y-axis. At this point, the value of x is zero. To find the y-intercept, you'll substitute 0 for x in your equation, and solve for y. This gives you the specific point at which the line reaches the vertical axis.
For the given equation:

• Set x = 0, and substitute into the equation: \(3y=6(0)+3\).
• Simplifying this gives \(3y=3\).
• Solve for y, resulting in \(y=1\).

Hence, the y-intercept is (0,1). By finding the y-intercept, you learn the starting height on the y-axis before your line moves towards the other points. It's like your line's starting block.

linear equations

Linear equations are mathematical expressions that rarely stray from straight line graphs. They're in the form \(ax + by = c\). The constants 'a', 'b', and 'c' are numbers that dictate the slope and position of the line. A linear equation involves variables having only power 1.
In the context of our exercise, you have the linear equation \(3y = 6x + 3\). Here:

• x and y are the variables representing coordinates.
• The equation is already in a form that can be rearranged and worked with to find intercepts.
• Solutions for x and y can be plotted on a graph to produce a straight line.

Understanding how to navigate linear equations helps you sketch graphs, identify points, and calculate distances on a coordinate plane. They're fundamental in algebra, drawing a simple bridge between numbers and graphs.