Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x-3 y=-1 \\ 10 x+y=11 \end{array}\right. $$

Short answer

The solution is x = 1, y = 1.

Step-by-step solution

- Write down the equations

Start with the given system of equations:\[\begin{cases}2x - 3y = -1 \ 10x + y = 11\end{cases}\]

- Eliminate one variable

First, solve the second equation for y:\[y = 11 - 10x\]Now substitute this expression for y in the first equation:\[2x - 3(11 - 10x) = -1\]

- Simplify and solve for x

Distribute and combine like terms:\[2x - 33 + 30x = -1\]\[32x - 33 = -1\]Add 33 to both sides:\[32x = 32\]Divide both sides by 32:\[x = 1\]

- Solve for y

Substitute x = 1 back into the equation y = 11 - 10x:\[y = 11 - 10(1)\]\[y = 11 - 10 = 1\]

- Check the solutions

Verify the solutions by substituting x = 1 and y = 1 back into the original equations:\[\begin{cases}2(1) - 3(1) = -1 & \text{True}\ 10(1) + 1 = 11 & \text{True}\end{cases}\]Both equations hold true, so the solution is correct.

Key concepts

solving linear equations

Linear equations are mathematical expressions that graph as straight lines on the coordinate plane. They follow the general form: \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Typically, a system of linear equations includes two or more equations that share the same set of unknowns. The goal is to find the values of these unknowns, making all equations true simultaneously.

In our exercise, we have two linear equations:
\begin{cases}2x - 3y = -1 ewline \ 10x + y = 11 ewline\right.Combining these, we introduce methods to solve linear systems, which one approach is the 'Substitution Method'.

substitution method

The substitution method commonly used for solving a system of linear equations involves substituting one variable for an equivalent expression involving the other variable.

The approach consists of these steps:

• Step 1: Solve one equation for one variable.
• Step 2: Substitute this solution in the other equation.
• Step 3: Solve the resulting single-variable equation.
• Step 4: Substitute back the found value to obtain the other variable.

For our system, starting by solving the second equation
'10x + y = 11' for \(y\):
\(y = 11 - 10x\)

Next, substitute \(y = 11 - 10x\) into the first equation to eliminate \(y\):
\(2x - 3(11 - 10x) = -1\)
Combine like terms and solve for \(x\). In our case:
\(32x = 32\)
So,\( x = 1\). Substituting \(x = 1\) back into \(y = 11 - 10x\) gives \(y\) as well, \(y = 1\).

consistent systems

When discussing systems of equations, we often categorize them based on the number of solutions:

• Consistent systems: These have at least one solution, meaning the lines intersect at one or multiple points.
• Inconsistent systems: These have no solutions and represent parallel lines that never meet.
• Dependent systems: These have infinite solutions, representing the same line (i.e., overlapping lines).

To check a system's consistency, look at the solutions. A consistent system will have at least one valid solution.
In our equation system:
1. The solution \(x = 1\) and \(y = 1\) makes both equations true when substituted back.
Thus, our system is consistent with a single unique solution: \((1, 1)\).