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Does the system of equations have no solution, one solution, or infinitely many solutions? \( \begin{array}{l}2 x+y=2 \\ y=-2 x-1\end{array} \) The system has Choose...

Ask by Ramirez Pearson. in the United States
Jan 22,2025

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The system has **no solution**.

Solución

To determine whether the given system of equations has no solution, one solution, or infinitely many solutions, let's analyze and solve the system: \[ \begin{cases} 2x + y = 2 \quad \text{(1)} \\ y = -2x - 1 \quad \text{(2)} \end{cases} \] **Step 1: Substitute Equation (2) into Equation (1)** Since Equation (2) expresses \( y \) in terms of \( x \), we can substitute \( y = -2x - 1 \) into Equation (1): \[ 2x + (-2x - 1) = 2 \] **Simplify the Equation:** \[ 2x - 2x - 1 = 2 \] \[ 0x - 1 = 2 \] \[ -1 = 2 \] **Step 2: Analyze the Result** The equation \( -1 = 2 \) is a contradiction—it's a statement that is never true. This means there's no value of \( x \) and \( y \) that can satisfy both equations simultaneously. **Conclusion:** The system of equations has **no solution**. **Summary:** - **No Solution:** The lines represented by the equations are parallel and never intersect. - **One Solution:** The lines intersect at a single point. - **Infinitely Many Solutions:** The equations represent the same line, so every point on the line is a solution. In this case, since we've arrived at a contradiction, the system has **no solution**. **Answer:** The system has **no solution**.

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Beyond the Answer

To determine the nature of the solutions for the given system of equations, we can graph the lines represented by each equation. The first equation, \(2x + y = 2\), can be rearranged to \(y = -2x + 2\), which has a slope of -2. The second equation, \(y = -2x - 1\), also has a slope of -2 but intersects the y-axis at -1. Since these two lines have the same slope but different y-intercepts, they are parallel and will never intersect—meaning this system has no solution. If you're ever confused about the relationship between two linear equations, just graph them! If they cross at a point, you've got one solution. If they're the same line, congrats—infinitely many solutions! But if they’re parallel, like here, then you can confidently say there's no solution!

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