Linear Equations with Two Variables: false statements and infinite solutions

To solve this system of equations, we'll determine if the equations are equivalent, which would imply infinite solutions.

First, observe the two equations:

• Equation 1: $5x - y = 0$
• Equation 2: $10x - 2y = 0$

Now, let's simplify Equation 2 by dividing every term by 2:

$\frac{10x}{2} - \frac{2y}{2} = \frac{0}{2}$

This simplifies to:

$5x - y = 0$

We see that simplifying the second equation results in the first equation:

Both equations are indeed the same, $5x - y = 0$.

Therefore, these two equations represent the same line in a coordinate plane, meaning they have an infinite number of solutions along this line.

There are infinite solutions.

To solve this system of equations, let's analyze the given equations:

• The first equation is $x + y = 0$.
• The second equation is $x + y = 10$.

Notice that the left-hand side of both equations is the same, $x + y$, but the right-hand sides are different: 0 and 10, respectively.

This means that there is no possible way for $x + y$ to equal both 0 and 10 at the same time. Hence, the equations contradict each other, and no pair $(x, y)$ can satisfy both equations simultaneously.

As a result, the system of equations is inconsistent. Therefore, the correct solution is that there is no solution to the system, which corresponds to choice No solution.

Therefore, the final solution to the problem is No solution.

We start by analyzing the given system of equations:

The first equation is $2x - 2y = 10$.

The second equation is $4x - 4y = 32$.

To determine the relationship between these two lines, let's simplify both equations.

1. Simplify the first equation:
Divide every term by 2:
$x - y = 5$.

2. Simplify the second equation:
Divide every term by 4:
$x - y = 8$.

Notice that after simplification, we have:

• First equation: $x - y = 5$
• Second equation: $x - y = 8$

Upon comparison, both equations simplify to lines with the same slope but different intercepts. Therefore, they represent two parallel lines that do not intersect.

Consequently, the system of equations has no solution since parallel lines never meet.

Therefore, the correct answer is: No solution.

To solve this system of equations, let's first look at the given equations:
Equation 1: $x - y = 8$
Equation 2: $2x - 2y = 16$

Let's simplify the second equation. We can divide the entire equation by 2:

$\frac{2x - 2y}{2} = \frac{16}{2}$

This simplifies to:
$x - y = 8$

We can see now that both equations are identical:
1. $x - y = 8$
2. $x - y = 8$

Since both equations represent the same line, every point that is a solution to the first equation is also a solution to the second equation. This means that there are infinitely many solutions, as every point on the line $x - y = 8$ is a solution.

Therefore, the system of equations has infinite solutions.

To solve this problem, let's apply the elimination method:

• Step 1: Write down the given equations: $3x - y = -10$ and $9x - 3y = -15$.
• Step 2: Observe that the second equation is 3 times the first equation. Multiply the first equation by 3 for alignment: $3(3x - y) = 3(-10)$ becomes $9x - 3y = -30$.
• Step 3: Compare the two resulting equations: $9x - 3y = -30$ and $9x - 3y = -15$.
• Step 4: Notice these equations suggest a contradiction as the left-hand side is the same ($9x - 3y$), but the right-hand sides are different ($-30$ vs. $-15$).

As the comparison shows a contradiction, these lines are parallel and do not intersect.
Therefore, the system of equations has no solution.

To solve this system of equations, we need to determine the relationship between the two equations. The given equations are:

$3x - y = -5$

$9x - 3y = -15$

Let's examine the second equation:

Notice that if we multiply the first equation by 3, we obtain:

$3(3x - y) = 3(-5)$

which simplifies to:

$9x - 3y = -15$

This is exactly the same as the second given equation. Thus, the second equation is a multiple of the first equation, indicating that they represent the same line in the coordinate plane.

When two equations represent the same line, any point on this line will satisfy both equations. Therefore, there are infinitely many solutions to this system. That is, there are infinitely many points $(x, y)$ that can satisfy both equations.

Therefore, the solution to the problem is Infinite solutions.

To solve this problem, we'll follow these steps:

• Step 1: Express one equation in terms of a single variable.
• Step 2: Substitute into the other equation.
• Step 3: Evaluate and determine the solution type.

Now, let's work through each step:
Step 1: Solve the first equation for $x$:
$\frac{x}{2} + y = 3 \quad \Rightarrow \quad x = 6 - 2y$.

Step 2: Substitute $x = 6 - 2y$ into the second equation:
$x + 2y = 6 \quad \Rightarrow \quad (6 - 2y) + 2y = 6$.

Step 3: Simplifying equation:
$6 - 2y + 2y = 6 \quad \Rightarrow \quad 6 = 6$.

This equation $6 = 6$ is always true, indicating that both equations are dependent and represent the same line.

Therefore, the system has infinite solutions.