Solve by graphing.
a)
[tex]$\begin{array}{l}
x+y=4 \
2 x+2 y=10\end{array}$[/tex]
Answer (2)
Rewrite the equations in slope-intercept form: y = − x + 4 and y = − x + 5 .
Observe that the slopes are the same ( − 1 ) and the y-intercepts are different ( 4 and 5 ).
Conclude that the lines are parallel and do not intersect.
State that the system has no solution: no solution .
Explanation
Understanding the Problem We are given a system of two linear equations:
x + y = 4 2 x + 2 y = 10
We need to solve this system by graphing. This means we will plot both lines on a coordinate plane and find their intersection point, which represents the solution to the system.
Rewriting the First Equation First, let's rewrite each equation in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept.
For the first equation, x + y = 4 , we subtract x from both sides to get:
y = − x + 4
So, the slope of the first line is m 1 = − 1 and the y-intercept is b 1 = 4 .
Rewriting the Second Equation For the second equation, 2 x + 2 y = 10 , we first divide both sides by 2 to simplify:
x + y = 5
Now, subtract x from both sides to get:
y = − x + 5
So, the slope of the second line is m 2 = − 1 and the y-intercept is b 2 = 5 .
Analyzing the Equations Now we can analyze the equations. Notice that both lines have the same slope ( m 1 = m 2 = − 1 ), which means they are parallel. However, they have different y-intercepts ( b 1 = 4 and b 2 = 5 ), so they are different lines. Parallel lines never intersect.
Determining the Solution Since the two lines are parallel and do not intersect, the system of equations has no solution. This means there are no values of x and y that satisfy both equations simultaneously.
Examples
Imagine you're trying to meet a friend. You both start at different locations and walk at the same speed in the same direction. If you start at different points, you'll never meet! This is like our system of equations with parallel lines – they have the same slope (speed and direction) but different starting points (y-intercepts), so they never intersect (meet). Understanding systems of equations helps in scenarios like coordinating movements, planning resource allocations, or optimizing processes where multiple constraints must be satisfied simultaneously.
To solve the system of equations by graphing, we found that both lines have the same slope but different y-intercepts, indicating that they are parallel. As parallel lines do not intersect, the system has no solution. Therefore, we conclude that there is no solution to this system of equations.
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