Questions: Solve the following system of linear equations by substitution and determine whether the system has one solution, no solution, or an infinite number of solutions. If the system has one solution, find the solution. -1.5 x + 0.5 y = -7 1.5 x - 0.5 y = 7

Solve the following system of linear equations by substitution and determine whether the system has one solution, no solution, or an infinite number of solutions. If the system has one solution, find the solution. -1.5 x + 0.5 y = -7 1.5 x - 0.5 y = 7
Transcript text: Solve the following system of linear equations by substitution and determine whether the system has one solution, no solution, or an infinite number of solutions. If the system has one solution, find the solution. \[ \left\{\begin{array}{c} -1.5 x+0.5 y=-7 \\ 1.5 x-0.5 y=7 \end{array}\right. \]
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Solution

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Solution Steps

To solve the given system of linear equations by substitution, we first solve one of the equations for one variable in terms of the other. Then, substitute this expression into the other equation to find the value of one variable. Finally, substitute back to find the value of the other variable. Check if the system has one solution, no solution, or an infinite number of solutions.

To solve the given system of linear equations by substitution, we will follow these steps:

Step 1: Solve One Equation for One Variable

We start by solving one of the equations for one of the variables. Let's solve the first equation for \( y \).

The first equation is: \[ -1.5x + 0.5y = -7 \]

Rearrange it to solve for \( y \): \[ 0.5y = 1.5x - 7 \]

Divide every term by 0.5 to isolate \( y \): \[ y = 3x - 14 \]

Step 2: Substitute the Expression into the Other Equation

Now, substitute the expression for \( y \) from Step 1 into the second equation.

The second equation is: \[ 1.5x - 0.5y = 7 \]

Substitute \( y = 3x - 14 \) into this equation: \[ 1.5x - 0.5(3x - 14) = 7 \]

Distribute the \(-0.5\): \[ 1.5x - 1.5x + 7 = 7 \]

Step 3: Simplify and Determine the Type of Solution

Simplify the equation: \[ 0x + 7 = 7 \]

This simplifies to: \[ 7 = 7 \]

Since this is a true statement and there are no variables left, it indicates that the system has an infinite number of solutions. The two equations are dependent, meaning they represent the same line.

Final Answer

The system of equations has an \(\boxed{\text{infinite number of solutions}}\).

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