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Problem 1

# Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \begin{aligned} x-3 y &=-1 \\ 4 x+3 y &=11 \end{aligned}

Expert verified
The given system of linear equations has one and only one solution, which is $$(x,y) = (2,1)$$.
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## Step 1: Combine the equations to eliminate a variable

: We will first try to eliminate the variable y from both equations. To do this, add both equations: $x - 3y = -1$ $4x + 3y = 11$ Add both equations: $(x - 3y) + (4x + 3y) = -1 + 11$

## Step 2: Simplify the combined equation

: Now, simplify the combined equation to find the value of x: $1x + 4x = 10$ $5x = 10$ Now, divide both sides by 5: $\frac{5x}{5} = \frac{10}{5}$ $x = 2$

## Step 3: Substitute the value of variable x in one of the equations

: Now that we have the value of x, we can substitute this value into either of the equations to find the value of y. We'll use the first equation: $x - 3y = -1$ $2 - 3y = -1$

## Step 4: Solve for y

: Now, solve for y: $3y = 2 + 1$ $3y = 3$ Divide both sides by 3: $\frac{3y}{3} = \frac{3}{3}$ $y = 1$

## Step 5: Finalize the results

: We have found unique values for both x and y: $x = 2$ $y = 1$ Since we have found unique values for both variables, the given system of linear equations has one and only one solution. The single and unique solution to the system is: $(x,y) = (2,1)$

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