QuestionSolve the system of equations. If the system has no solution, say that it is inconsistent. Graph the lines of the system.
Select the correct choice below and, if necessary, fill in any answer boxes within your choice.
A. The solution is and . (Type integers or simplified fractions.)
B. There are infinitely many solutions. Using ordered pairs, they can be expressed as , y is any real number
(Simplify your answer. Type an expression using y as the variable as needed.)
C. The system is inconsistent.
Studdy Solution
STEP 1
1. We are given a system of two linear equations.
2. The system can have one solution, infinitely many solutions, or no solution.
3. We will use algebraic methods to determine the solution and graph the lines to visualize the system.
STEP 2
1. Analyze the system of equations.
2. Use algebraic methods to solve the system.
3. Determine the nature of the solution.
4. Graph the lines of the system.
STEP 3
Consider the given system of equations:
\begin{align*}
2x + y &= 4 \quad \text{(Equation 1)} \\
4x + 2y &= 3 \quad \text{(Equation 2)}
\end{align*}
STEP 4
To solve the system, let's use the method of elimination. First, we'll manipulate the equations to eliminate one of the variables.
Multiply Equation 1 by 2 to align the coefficients of :
\begin{align*}
4x + 2y &= 8 \quad \text{(Modified Equation 1)} \\
4x + 2y &= 3 \quad \text{(Equation 2)}
\end{align*}
Subtract Equation 2 from Modified Equation 1:
STEP 5
Since we arrived at a contradiction (0 = 5), the system of equations has no solution.
STEP 6
Given the contradiction, the system is inconsistent. Therefore, the correct choice is:
C. The system is inconsistent.
STEP 7
To graph the lines, rewrite each equation in slope-intercept form ():
Equation 1:
Equation 2:
Graph these lines on the same coordinate plane. Notice that they are parallel lines with different intercepts, confirming that there is no intersection point and thus no solution.
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