$\begin{aligned}
y & =5 x-1 \\
-15 x-3 y & =3
\end{aligned}$
How many solutions does this linear system have?
A. one solution: $(0,-1)$
B. one solution: $(1,4)$
C. no solution
D. infinite number of solutions
Answer (1)
The given system of linear equations has one solution. By solving the system, we find that the solution is ( 0 , − 1 ) . Therefore, the system has a unique solution. one solution: ( 0 , − 1 )
Explanation
Analyze the problem We are given a system of two linear equations:
Equation 1: y = 5 x − 1 Equation 2: − 15 x − 3 y = 3
Our goal is to determine the number of solutions this system has. We can start by rewriting both equations in the standard form A x + B y = C .
Rewrite the equations Equation 1 can be rewritten as:
− 5 x + y = − 1
Equation 2 can be rewritten as:
− 15 x − 3 y = 3
We can simplify Equation 2 by dividing both sides by -3:
5 x + y = − 1
Solve for y Now we have the following system of equations:
− 5 x + y = − 1 5 x + y = − 1
We can solve this system by adding the two equations to eliminate x :
( − 5 x + y ) + ( 5 x + y ) = − 1 + ( − 1 ) 2 y = − 2 y = − 1
Solve for x Now that we have the value of y , we can substitute it back into either equation to solve for x . Let's use the first equation:
− 5 x + ( − 1 ) = − 1 − 5 x = 0 x = 0
Determine the number of solutions So, the solution to the system is ( 0 , − 1 ) . Since we found a unique solution, the system has one solution.
Alternative Analysis Alternatively, we can analyze the equations after rewriting them:
− 5 x + y = − 1 5 x + y = − 1
These are two distinct lines that are not parallel, because their slopes are different. The first equation can be written as y = 5 x − 1 , which has a slope of 5. The second equation can be written as y = − 5 x − 1 , which has a slope of -5. Since the slopes are different, the lines intersect at one point, meaning there is exactly one solution.
Examples
Systems of linear equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, balancing chemical equations, and analyzing electrical circuits. For example, a company might use a system of equations to determine how many units of two different products they need to sell to reach a specific revenue target, considering the cost of production and the selling price of each product. By solving the system, they can find the exact quantities needed to meet their goal.
