Graph the equations:
\[
\begin{array}{l}
y=3 x+1 \\
y=-x+1
\end{array}
\]
On the desmos/graphing calculator and sketch the graph. Make sure to have two points and use a ruler.
Answer (2)
Find two points on the line y = 3 x + 1 : ( 0 , 1 ) and ( 1 , 4 ) .
Find two points on the line y = − x + 1 : ( 0 , 1 ) and ( 1 , 0 ) .
Sketch both lines on a coordinate plane using these points.
The lines intersect at ( 0 , 1 ) .
Explanation
Understanding the Equations We are given two linear equations, y = 3 x + 1 and y = − x + 1 , and our goal is to graph them. We'll find two points on each line to accurately sketch the graph.
Finding Points on the First Line For the first equation, y = 3 x + 1 , let's find two points. If x = 0 , then y = 3 ( 0 ) + 1 = 1 . So, the point is ( 0 , 1 ) . If x = 1 , then y = 3 ( 1 ) + 1 = 4 . So, the point is ( 1 , 4 ) .
Finding Points on the Second Line For the second equation, y = − x + 1 , let's find two points. If x = 0 , then y = − ( 0 ) + 1 = 1 . So, the point is ( 0 , 1 ) . If x = 1 , then y = − ( 1 ) + 1 = 0 . So, the point is ( 1 , 0 ) .
Sketching the Graph Now we have two points for each line: ( 0 , 1 ) and ( 1 , 4 ) for y = 3 x + 1 , and ( 0 , 1 ) and ( 1 , 0 ) for y = − x + 1 . We can sketch these lines on a coordinate plane, making sure to use a ruler to draw straight lines through the points. Notice that both lines intersect at the point ( 0 , 1 ) .
Examples
Graphing linear equations is a fundamental skill in algebra and has many real-world applications. For example, if you are tracking the distance you travel over time at a constant speed, you can represent this relationship with a linear equation and graph it to visualize your progress. Similarly, if you are comparing two different phone plans with different monthly fees and per-minute charges, you can graph the cost of each plan as a function of the number of minutes used to determine which plan is more cost-effective for your usage.
To graph the equations y = 3 x + 1 and y = − x + 1 , you find points on each line such as ( 0 , 1 ) and ( 1 , 4 ) for the first and ( 0 , 1 ) and ( 1 , 0 ) for the second. These points can be plotted on a coordinate plane and connected with straight lines using a ruler. The lines intersect at the point ( 0 , 1 ) .
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