What is the solution of this system of linear equations?
[tex]\begin{array}{l}
3 y=\frac{3}{2} x+6 \\
\frac{1}{2} y-\frac{1}{4} x=3
\end{array}[/tex]
A. (3,6)
B. (2,1)
C. no solution
D. infinite number of solutions
Answer (2)
Rewrite the equations in slope-intercept form.
Compare the slopes and y-intercepts.
If the slopes are the same but the y-intercepts are different, the system has no solution.
Therefore, the system has no solution .
Explanation
Analyze the problem We are given the following system of linear equations:
3 y = 2 3 x + 6
2 1 y − 4 1 x = 3
Our goal is to determine the solution of this system.
Rewrite equations First, let's rewrite both equations in slope-intercept form ( y = m x + b ).
Equation 1: 3 y = 2 3 x + 6 . Divide both sides by 3 to get y = 2 1 x + 2 .
Equation 2: 2 1 y − 4 1 x = 3 . Multiply both sides by 2 to get y − 2 1 x = 6 . Add 2 1 x to both sides to get y = 2 1 x + 6 .
Compare slopes and intercepts Now we have the two equations in slope-intercept form:
y = 2 1 x + 2
y = 2 1 x + 6
We can see that the slopes of both lines are the same ( m = 2 1 ), but the y-intercepts are different (2 and 6). This means the lines are parallel and will never intersect.
Conclusion Since the lines are parallel and do not intersect, the system of equations has no solution.
Examples
Systems of linear equations are used in various real-world applications, such as determining the break-even point for a business, optimizing resource allocation, and modeling supply and demand in economics. In this case, if the two equations represented cost and revenue, the lack of a solution would indicate that the cost will always be higher than the revenue, meaning the business will never be profitable.
The system of equations has no solution because the lines represented by the equations are parallel, having the same slope and different y-intercepts.
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