Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}4 b+3 m=3 \\ 3 b+11 m=13\end{array}\right.$$

Short answer

The solution for the system of equations are \(b = \frac{1}{35}\) and \(m = \frac{43}{35}\). And by substituting these values into the original equations, it can be verified that the system is consistent.

Step-by-step solution

Multiply the equations

In order to eliminate one variable, multiply first equation by 3 and the second equation by 4 to get: \(12b+9m=9\) and \(12b+44m=52\).

Subtract Equations

Subtract the first equation from the second to get \(35m=43\).

Solve for m

Divide both sides of the equation by 35 to solve for m: \(m = \frac{43}{35}\).

Substitute m into original equation

Substitute the value of m in one of the original equations to get value of b, let's use \( 4b + 3*\frac{43}{35} = 3\).

Simplify and solve for b

Solving for b gives, \(b = \frac{1}{35}\).

Check for consistency

Now substitute the obtained values of b and m into the original equation to check if the system is consistent.

Prove

Substitution of b and m into the original eqs should result in true statements where the left hand side equals the right hand side, showing that the system is consistent.

Key concepts

Elimination Method

The elimination method is a technique used to solve systems of linear equations. It's an excellent tool that allows you to eliminate one of the variables by adding or subtracting the equations. This helps simplify the problem so that you can easily find the values of the remaining variables.

Here’s how it works with a system of two equations:

• First, manipulate the equations to align the coefficients of one variable in both equations, which may involve multiplying the entire equation by a constant.
• Next, add or subtract the equations to eliminate one variable, allowing you to solve for the other.
• Finally, substitute the known variable's value back into one of the original equations to find the remaining variable.

In our exercise, we used this method by adjusting the coefficients of the variable 'b', allowing us to solve the system more straightforwardly. Remember that the key here is to eliminate strategically, so you end up with a solvable equation.

Consistent System

A system of linear equations is consistent if there is at least one set of variable values that satisfy all equations in the system. For instance, a system is consistent if, when solved, it yields a true statement such as 0 = 0.

There are two types of consistent systems:

• Independent Consistent System: There is only one unique set of solutions. For example, two lines intersecting at a single point.
• Dependent Consistent System: There are infinitely many solutions because the equations coincide (i.e., they represent the same line).

In contrast, if the system has no solutions, it is called inconsistent. This would mean the equations contradict each other, often represented by parallel lines that never intersect.

In our scenario, upon checking the solutions for 'b' and 'm' using substitution, we verified that the system is consistent because these values satisfy both equations.

Substitution Method

The substitution method is an alternative to the elimination method and involves substituting one variable's expression into another equation. This method can be particularly useful when one of the variables is already isolated or can be easily isolated. Here's how it works:

• First, solve one of the equations for one of its variables.
• Next, substitute this expression into the other equation, which transforms it into an equation with a single variable.
• Finally, solve this new equation for the remaining variable and use its value to find the first variable by substitution.

In our step-by-step solution, the substitution method was used after finding 'm' to identify 'b'. We substituted the value of 'm' into one of the original equations to retrieve 'b', showcasing the flexibility and effectiveness of this technique.

Linear Equations

Linear equations are equations of the first order. This means they have no exponents higher than one, and they graph as straight lines in a coordinate plane. Linear equations generally take the form of: \[ ax + by = c \] Where 'a', 'b', and 'c' are constants. The solution to a system of linear equations is the point where their graphs intersect.

In any linear system, there can be:

• Exactly one solution (the lines intersect at one point).
• Infinitely many solutions (the lines coincide, representing the same line).
• No solutions (the lines are parallel and do not intersect).

Understanding how to solve linear systems is crucial for tackling real-world problems, from determining supply and demand in economics to routing networks in computer science.