What is the equation of the following line written (1,1) 2x – y – 1 = 0
The Correct Answer and Explanation is:
Of course. Here is the solution and a detailed explanation.
Correct Answer:
The equation of the line is 2x – y – 1 = 0.
Explanation
To determine the equation of a line from its graph, one must first find its fundamental properties: the slope and a point on the line. The equation can then be constructed using these properties.
Step 1: Finding the Slope of the Line
The slope, often denoted by the variable m, measures the steepness of a line. It is calculated as the “rise” (the vertical change) divided by the “run” (the horizontal change) between any two distinct points on the line. The formula for the slope is:
m = (y₂ – y₁) / (x₂ – x₁)
From the graph, we can identify two points where the line intersects the grid perfectly:
- The point explicitly labeled: (1, 1)
- The point where the line crosses the y-axis (the y-intercept): (0, -1)
Let (x₁, y₁) be (0, -1) and (x₂, y₂) be (1, 1). Now, substitute these values into the slope formula:
m = (1 – (-1)) / (1 – 0)
m = (1 + 1) / 1
m = 2 / 1
m = 2
The slope of the line is 2.
Step 2: Using the Slope-Intercept Form
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
From Step 1, we found the slope m = 2.
The y-intercept is the point where the line crosses the y-axis. As identified from the graph, this occurs at (0, -1), so b = -1.
Substituting these values into the slope-intercept form gives:
y = 2x + (-1)
y = 2x – 1
Step 3: Converting to Standard Form
The answer options are provided in the standard form of a linear equation, which is Ax + By + C = 0. To convert our equation (y = 2x – 1) into this format, we rearrange the terms so that they are all on one side of the equation.
Starting with y = 2x – 1, we can subtract y from both sides:
0 = 2x – y – 1
By convention, this is written as:
2x – y – 1 = 0
This matches the given option. To verify, we can substitute the point (1, 1) into this equation:
2(1) – (1) – 1 = 0
2 – 1 – 1 = 0
0 = 0
Since the equation holds true, it correctly represents the graphed line.thumb_upthumb_down
