Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}\frac{x}{4}+\frac{y}{6}=1 \\\ x-y=3\end{array}\right.$$

Short answer

The solution to the system of equations is \(x = 6\), \(y = 3\). The system is consistent as it has a unique solution.

Step-by-step solution

Eliminate one of the variables

In the given equations, the second equation is easier to handle because it is already in a form where the coefficients of \(x\) and \(y\) are integers. We can multiply the first equation by 4 to get rid of the denominator. The new system of equations now becomes \[4* (\frac{x}{4}+\frac{y}{6})=4*1\]i.e. \(x+ \frac{2}{3}y =4\) and \(x - y = 3\)

Substitution

Now we make the coefficients of y in both the equations same (or opposite) to easily cancel it. Multiply the first equation by 3 and the second equation by 2. Then subtract the second equation from the first one. The new system of equation after substitution becomes: \[3*(x + \frac{2}{3}y)= 3*4\] and \[2*(x - y)= 2*3\] Therefore we get, \(3x + 2y = 12\) and\(2x - 2y = 6\). The difference of these two equations is: \(3x - 2x = 12 - 6\), which simplifies to \(x = 6\)

Solving For Second Variable

Now we substitute \(x = 6\) into the second given equation \(x - y = 3\) to get \(6- y = 3\), which simplifies to \(y = 3\)

Determining Consistency

Since we have unique solutions for \(x\) and \(y\), we can conclude that this system of equations is consistent

Key concepts

Elimination Method

The elimination method is an efficient technique employed in solving systems of equations. It involves manipulating the equations to eliminate one variable, making it simpler to isolate the other. To understand this better, envision you have two balancing scales, each with different items but one common item on both. By suitably adding or subtracting the weight on one scale, you can balance out the common item, effectively 'eliminating' it and isolating the other items.

When applying this to algebra, you would adjust the coefficients of one variable across different equations until they match or are exact opposites. By then adding or subtracting the equations, one variable is canceled out, thereby simplifying the system to a single equation with one variable. This streamlined process is highly effective, especially when the equations are not easily the manipulatable for algebraic substitution.

Consistent Systems

A system of equations is deemed consistent when it possesses at least one set of solutions. Imagine inviting mutual friends to a party; a consistent situation would be where at least one friend can attend. In the realm of algebra, this equates to a point or multiple points of intersection between lines, indicating where the equations hold true simultaneously. Consistency comes in two flavors:

• Consistently Independent: This occurs when there is exactly one solution, represented graphically as two lines intersecting at a single point.
• Consistently Dependent: Infinite solutions exist here, showcased with two lines perfectly superimposed over each other on the graph.

In contrast, an inconsistent system is where no solutions can be found since the lines are parallel and never meet—like two friends with clashing schedules unable to attend your party together.

System of Linear Equations

A system of linear equations is a collection of two or more linear equations with the same variables. Picture a map with various routes to get to a treasure chest; each linear equation is like a potential path, and the solution of the system is the point where the paths might intersect, leading you to the treasure. This system can be visualized graphically with straight lines on a coordinate plane, each line representing a possible path or equation.

Such systems are classified based on the number of solutions they offer—no solution, a single solution, or infinitely many solutions. Solving these systems can be approached through different methods, such as graphing, substitution, or elimination. The elimination method, which we previously discussed, is particularly useful when the system involves equations not easily rearranged for substitution.

Algebraic Substitution

Algebraic substitution is akin to swapping ingredients in a recipe to alter the flavor but still achieving a delicious meal. In algebra, it means replacing one variable with an equivalent expression derived from another equation within the system. This substitution simplifies the system to one equation with one variable.

This method is particularly handy when one equation in the system is already solved for a variable or can be manipulated with ease to reach that form. Once a variable is expressed in terms of another, you substitute this expression into the other equation, converting the system's complexity down to a simpler equation that is straightforward to solve. Whether it's finding the right balance in a recipe or in equations, substitution is all about finding the perfect match to solve a problem.