Math / Algebra

QuestionQuestion Which of the following systems of equations has no solution?
$\begin{array}{l} x-2 y=2 \\ x-2 y=4 \end{array}$
$\begin{array}{r} x+2 y=7 \\ x=y \end{array}$
$\begin{array}{l} x-y=0 \\ x+y=1 \end{array}$
$\begin{aligned} x-y & =1 \\ x & =2 y \end{aligned}$

Studdy Solution

STEP 1

1. A system of equations has no solution if the equations represent parallel lines that never intersect.
2. We will analyze each system of equations to determine if they are parallel or inconsistent.

STEP 2

1. Analyze the first system of equations.
2. Analyze the second system of equations.
3. Analyze the third system of equations.
4. Analyze the fourth system of equations.
5. Determine which system has no solution.

STEP 3

Analyze the first system of equations:
\begin{align} x - 2y &= 2 \\ x - 2y &= 4 \end{align}
Both equations have the same left-hand side but different right-hand sides. This indicates that they are parallel lines with no intersection.

STEP 4

Analyze the second system of equations:
\begin{align} x + 2y &= 7 \\ x &= y \end{align}
Substitute $x = y$ into the first equation:
$y + 2y = 7 \implies 3y = 7 \implies y = \frac{7}{3}$
Substitute back to find $x$ :
$x = \frac{7}{3}$
This system has a solution $(x, y) = \left(\frac{7}{3}, \frac{7}{3}\right)$ .

STEP 5

Analyze the third system of equations:
\begin{align} x - y &= 0 \\ x + y &= 1 \end{align}
Solve the first equation for $x$ :
$x = y$
Substitute into the second equation:
$y + y = 1 \implies 2y = 1 \implies y = \frac{1}{2}$
Substitute back to find $x$ :
$x = \frac{1}{2}$
This system has a solution $(x, y) = \left(\frac{1}{2}, \frac{1}{2}\right)$ .

STEP 6

Analyze the fourth system of equations:
\begin{align} x - y &= 1 \\ x &= 2y \end{align}
Substitute $x = 2y$ into the first equation:
$2y - y = 1 \implies y = 1$
Substitute back to find $x$ :
$x = 2 \times 1 = 2$
This system has a solution $(x, y) = (2, 1)$ .

STEP 7

Determine which system has no solution:
The first system of equations:
\begin{align} x - 2y &= 2 \\ x - 2y &= 4 \end{align}
is the only one with no solution because the equations represent parallel lines.
The system with no solution is the first one.

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QuestionQuestion Which of the following systems of equations has no solution?
$\begin{array}{l} x-2 y=2 \\ x-2 y=4 \end{array}$
$\begin{array}{r} x+2 y=7 \\ x=y \end{array}$
$\begin{array}{l} x-y=0 \\ x+y=1 \end{array}$
$\begin{aligned} x-y & =1 \\ x & =2 y \end{aligned}$

Studdy Solution

STEP 1

1. A system of equations has no solution if the equations represent parallel lines that never intersect.
2. We will analyze each system of equations to determine if they are parallel or inconsistent.

STEP 2

1. Analyze the first system of equations.
2. Analyze the second system of equations.
3. Analyze the third system of equations.
4. Analyze the fourth system of equations.
5. Determine which system has no solution.

STEP 3

Analyze the first system of equations:
\begin{align} x - 2y &= 2 \\ x - 2y &= 4 \end{align}
Both equations have the same left-hand side but different right-hand sides. This indicates that they are parallel lines with no intersection.

STEP 4

STEP 5

STEP 6

Analyze the fourth system of equations:
\begin{align} x - y &= 1 \\ x &= 2y \end{align}
Substitute $x = 2y$ into the first equation:
$2y - y = 1 \implies y = 1$
Substitute back to find $x$ :
$x = 2 \times 1 = 2$
This system has a solution $(x, y) = (2, 1)$ .

STEP 7

Determine which system has no solution:
The first system of equations:
\begin{align} x - 2y &= 2 \\ x - 2y &= 4 \end{align}
is the only one with no solution because the equations represent parallel lines.
The system with no solution is the first one.

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Studdy Solution

STEP 1

1. A system of equations has no solution if the equations represent parallel lines that never intersect.2. We will analyze each system of equations to determine if they are parallel or inconsistent.

STEP 2

1. Analyze the first system of equations.2. Analyze the second system of equations.3. Analyze the third system of equations.4. Analyze the fourth system of equations.5. Determine which system has no solution.

STEP 3

Analyze the first system of equations:\begin{align*} x - 2y &= 2 \\ x - 2y &= 4 \end{align*}Both equations have the same left-hand side but different right-hand sides. This indicates that they are parallel lines with no intersection.

STEP 4

STEP 5

STEP 6

STEP 7

Determine which system has no solution:The first system of equations:\begin{align*} x - 2y &= 2 \\ x - 2y &= 4 \end{align*}is the only one with no solution because the equations represent parallel lines. The system with no solution is the first one.

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Studdy solves anything!

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. A system of equations has no solution if the equations represent parallel lines that never intersect.
2. We will analyze each system of equations to determine if they are parallel or inconsistent.

1. Analyze the first system of equations.
2. Analyze the second system of equations.
3. Analyze the third system of equations.
4. Analyze the fourth system of equations.
5. Determine which system has no solution.

Analyze the first system of equations:
\begin{align} x - 2y &= 2 \\ x - 2y &= 4 \end{align}
Both equations have the same left-hand side but different right-hand sides. This indicates that they are parallel lines with no intersection.

Determine which system has no solution:
The first system of equations:
\begin{align} x - 2y &= 2 \\ x - 2y &= 4 \end{align}
is the only one with no solution because the equations represent parallel lines.
The system with no solution is the first one.