Let's solve the system of equations given by:
- Equation 1: \( x = 4 \)
- Equation 2: \( y = 2x - 3 \)
Graphing the Equations
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Graph of \( x = 4 \):
- This equation represents a vertical line that crosses the x-axis at \( x = 4 \). The line is vertical and extends infinitely in both the positive and negative directions of the y-axis.
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Graph of \( y = 2x - 3 \):
- This is a linear equation in slope-intercept form \( y = mx + b \), where \( m = 2 \) (the slope) and \( b = -3 \) (the y-intercept).
- To graph this, we can find points. When \( x = 0 \), \( y = -3 \) (the y-intercept). When \( x = 4 \), \( y = 2(4) - 3 = 8 - 3 = 5 \).
- So we can plot the points (0, -3) and (4, 5) and draw a line through these points.
Finding the Intersection
To find how many solutions we are expecting and why, we look for the intersection of these two lines:
- Since \( x = 4 \) is a vertical line, it will intersect the line \( y = 2x - 3 \) at \( x = 4 \).
- To find the corresponding y-value, substitute \( x = 4 \) into the second equation: \[ y = 2(4) - 3 = 8 - 3 = 5 \]
- So the two lines intersect at the point \( (4, 5) \).
Conclusion on Solutions
Thus, there is one solution to this system of equations, and the solution is: \[ (4, 5) \]
Observations
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What do you notice about the line \( x = 4 \)?
The line \( x = 4 \) is a vertical line that represents all points where the x-coordinate is 4, regardless of the y-coordinate. -
What equations would make a horizontal line?
An equation of the form \( y = c \) where \( c \) is a constant will represent a horizontal line. For example, \( y = 3 \) represents a horizontal line that crosses the y-axis at \( y = 3 \) and extends indefinitely in the positive and negative directions of the x-axis.
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