Examine the system of equations. Do you think the lines will intersect? Explain.
[tex]
\begin{array}{l}
y=2 x-7 \\
y=x-7
\end{array}
[/tex]
Answer (2)
Set the two equations equal to each other: 2 x − 7 = x − 7 .
Solve for x : x = 0 .
Substitute x into one of the equations to find y : y = − 7 .
The lines intersect at the point ( 0 , − 7 ) , so the answer is yes, the lines intersect.
Explanation
Analyzing the System of Equations We are given a system of two linear equations:
y = 2 x − 7
y = x − 7
The question asks whether the lines represented by these equations will intersect. To determine this, we need to find out if there is a solution to this system of equations. If a solution exists, the lines intersect at that point.
Setting the Equations Equal To find the intersection point (if it exists), we can set the two equations equal to each other:
2 x − 7 = x − 7
Solving for x Now, we solve for x :
2 x − x = 7 − 7
x = 0
Solving for y Next, we substitute the value of x into either of the original equations to find the value of y . Let's use the second equation:
y = x − 7
y = 0 − 7
y = − 7
Conclusion We found a unique solution ( x , y ) = ( 0 , − 7 ) . This means the lines intersect at the point ( 0 , − 7 ) .
Examples
Imagine you are tracking the paths of two airplanes on a radar screen. Each airplane's path can be represented by a linear equation. If the equations intersect, it means the airplanes' paths cross at a certain point. Determining the intersection point helps air traffic controllers ensure the planes maintain a safe distance and avoid collisions. This is a practical application of solving systems of linear equations.
The two lines represented by the equations intersect at the point (0, -7). This is determined by setting the equations equal to each other, solving for x, and then finding y. Hence, there is a unique solution where the lines cross.
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