To solve the given linear equation [tex]\(y = 3x - 1\)[/tex], we can start by understanding its components and characteristics.
### Step-by-Step Breakdown:
1. Identify the Linear Equation:
The given equation is [tex]\(y = 3x - 1\)[/tex].
2. Understand the Structure:
- This is a linear equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where:
- [tex]\(m\)[/tex] is the slope (rate of change).
- [tex]\(b\)[/tex] is the y-intercept (the value of [tex]\(y\)[/tex] when [tex]\(x = 0\)[/tex]).
In this equation:
- The slope [tex]\(m\)[/tex] is 3.
- The y-intercept [tex]\(b\)[/tex] is -1.
3. Plotting the Equation on a Graph:
To visualize this linear equation on a graph:
- Begin at the y-intercept [tex]\(b = -1\)[/tex]. This means the graph crosses the y-axis at [tex]\((0, -1)\)[/tex].
- From this point, use the slope to determine other points. The slope [tex]\(3\)[/tex] means that for every 1 unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by 3 units.
4. Finding Specific Points:
- When [tex]\(x = 0\)[/tex]:
[tex]\[
y = 3(0) - 1 = -1
\][/tex]
So, the point is [tex]\((0, -1)\)[/tex].
- When [tex]\(x = 1\)[/tex]:
[tex]\[
y = 3(1) - 1 = 2
\][/tex]
So, the point is [tex]\((1, 2)\)[/tex].
- When [tex]\(x = -1\)[/tex]:
[tex]\[
y = 3(-1) - 1 = -4
\][/tex]
So, the point is [tex]\((-1, -4)\)[/tex].
5. Graphing the Line:
Plot the points [tex]\((0, -1)\)[/tex], [tex]\((1, 2)\)[/tex], and [tex]\((-1, -4)\)[/tex] on a coordinate plane. Draw a straight line through these points, extending in both directions to form the line that represents the equation [tex]\(y = 3x - 1\)[/tex].
### Summary:
Thus, the equation [tex]\(y = 3x - 1\)[/tex] describes a straight line with a slope of 3 and a y-intercept of -1. You can find various points on this line by selecting different values for [tex]\(x\)[/tex] and calculating the corresponding [tex]\(y\)[/tex], which will always follow the relationship [tex]\(y = 3x - 1\)[/tex].
