Solve the simultaneous equations
$\begin{array}{l}
y=-x+5 \text { and } y=x-1 . \\
x=\quad, y=
\end{array}$
We are given a system of two linear equations in two variables, \(x\) and \(y\):
\[ \begin{cases} y = -x + 5 \ y = x - 1 \end{cases} \]
Our objective is to find the point \((x, y)\) where these two lines intersect in the Cartesian plane. Since both equations are solved for \(y\), we can use the method of substitution to set the two expressions for \(y\) equal to each other.
Since both equations are equal to \(y\), we set their expressions equal to solve for \(x\):
\[-x + 5 = x - 1\]
To solve for \(x\), we first add \(x\) to both sides of the equation:
\[5 = 2x - 1\]
Next, add 1 to both sides:
\[6 = 2x\]
Finally, divide by 2:
\[x = 3\]
Now that we have found \(x = 3\), we can substitute this value back into either of the original equations to find \(y\). Let's use the second equation:
\[y = x - 1\]
Substituting \(x = 3\):
\[y = 3 - 1\]
\[y = 2\]
To ensure accuracy, we can verify this with the first equation:
\[y = -x + 5\]
Substituting \(x = 3\):
\[y = -3 + 5\]
\[y = 2\]
Both equations yield the same value for \(y\), confirming that the intersection point is \((3, 2)\).
Imagine you are tracking two moving objects on a map. Object A starts at a position and moves according to the rule \(y = -x + 5\), while Object B moves according to \(y = x - 1\). Finding the intersection of these lines is exactly how you would determine the precise moment and location where the two objects collide. Solving systems of equations like this is essential for navigation, traffic management, and computer graphics to determine when and where paths cross.