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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. \[ \begin{array}{l}4 x+3 y=7 \\ -12 x-10 y=-27\end{array} \] Answer Infinitely Many Solutions No Solutions

Ask by Stanley Henry. in the United States
Jan 22,2025

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Answer

The system has exactly one solution: \( x = -\frac{11}{4} \) and \( y = 6 \).

Solution

To determine whether the given system of equations has no solutions, infinitely many solutions, or exactly one solution, let's analyze the system: \[ \begin{cases} 4x + 3y = 7 \\ -12x - 10y = -27 \end{cases} \] **Step 1: Calculate the Determinant** First, consider the coefficient matrix: \[ \begin{bmatrix} 4 & 3 \\ -12 & -10 \end{bmatrix} \] The determinant (\(D\)) of this matrix is calculated as: \[ D = (4)(-10) - (-12)(3) = -40 + 36 = -4 \] Since the determinant \(D \neq 0\), the system has a **unique solution**. **Step 2: Solve the System** Alternatively, we can solve the system using substitution or elimination. 1. **Multiply the first equation by 3:** \[ 12x + 9y = 21 \] 2. **Add this to the second equation:** \[ (-12x - 10y) + (12x + 9y) = -27 + 21 \\ (-10y + 9y) = -6 \\ -y = -6 \\ y = 6 \] 3. **Substitute \(y = 6\) into the first equation:** \[ 4x + 3(6) = 7 \\ 4x + 18 = 7 \\ 4x = -11 \\ x = -\frac{11}{4} \] **Conclusion:** The system has exactly one solution: \[ x = -\frac{11}{4}, \quad y = 6 \] **Answer:** The system has exactly one solution.

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To determine the nature of the solution for the given system of equations, we can manipulate one equation to express its relationship with the other. If we multiply the first equation, \(4x + 3y = 7\), by 3, we get \(12x + 9y = 21\). Now, if we notice, the second equation \(-12x - 10y = -27\) can be compared with \(12x + 9y = 21\) when we solve for \(y\). The two equations do not coincide, indicating that they are parallel lines, thus confirming that there are no solutions. Revisiting systems of equations, you can often find that parallel lines have the same slope but different y-intercepts, leading to that no single point satisfies both equations. In contrast, coincident lines (exactly the same line) yield infinitely many solutions, while intersecting lines result in a unique solution. Always check the slopes and intercepts!

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