Enter the unknown coordinates so that each ordered pair is a solution to the linear equation represented by the graph.
Understand the Problem
The question is asking to determine unknown coordinates for several ordered pairs that satisfy the linear equation depicted in the provided graph. The graph represents a line, and the user needs to find the y-coordinates corresponding to given x-coordinates for each pair.
Answer
1. \((-3.5, -6)\) 2. \((0.5, 2)\) 3. \((-2.25, -3.5)\) 4. \((20, 41)\)
Answer for screen readers
The unknown coordinates are as follows:
- ((-3.5, -6))
- ((0.5, 2))
- ((-2.25, -3.5))
- ((20, 41))
Steps to Solve
- Identify the linear equation from the graph
Observe the given graph to determine the equation of the line. The slope and y-intercept are key. From examining the graph, a possible equation can be noted as:
$$ y = mx + b $$
- Determine the slope of the line
Calculate the slope (m) by choosing two points on the line and using the formula:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
For example, if we use the points (0, 1) and (2, 5), the slope would be:
$$ m = \frac{5 - 1}{2 - 0} = \frac{4}{2} = 2 $$
- Identify the y-intercept
The y-intercept (b) is the point where the line crosses the y-axis. From the graph, this appears to be at (0, 1), therefore:
$$ b = 1 $$
- Write the linear equation
With the slope and intercept determined, the linear equation of the line is:
$$ y = 2x + 1 $$
- Find unknown y-coordinates for the x-values
Now substitute the x-values from the ordered pairs into the linear equation to find the corresponding y-values.
- For the first pair ((-3.5, ?)):
$$ y = 2(-3.5) + 1 = -7 + 1 = -6 $$
- For the second pair ((0.5, ?)):
$$ y = 2(0.5) + 1 = 1 + 1 = 2 $$
- For the third pair ((?, -3.5)):
To find the corresponding x-value for ( y = -3.5 ):
$$ -3.5 = 2x + 1 \implies 2x = -3.5 - 1 \implies 2x = -4.5 \implies x = -2.25 $$
- For the fourth pair ((20, ?)):
$$ y = 2(20) + 1 = 40 + 1 = 41 $$
- Compile the coordinates
The ordered pairs based on the calculations are:
- ((-3.5, -6))
- ((0.5, 2))
- ((-2.25, -3.5))
- ((20, 41))
The unknown coordinates are as follows:
- ((-3.5, -6))
- ((0.5, 2))
- ((-2.25, -3.5))
- ((20, 41))
More Information
This problem demonstrates how to extract information from graphical representations of linear equations. Understanding how to derive y-coordinates using the equation of a line is crucial for solving more complex algebraic problems.
Tips
- Forgetting to change the signs while calculating coordinates.
- Misinterpreting the slope or y-intercept from the graph.
- Not correctly substituting values into the equation.
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