Question

Solve simultaneously. Check some by calculator. $$\begin{aligned} &\frac{m}{2}+\frac{n}{3}-3=0\\\ &\frac{n}{2}+\frac{m}{5}=\frac{23}{10} \end{aligned}$$

Short answer

The solution to the system is \(m = 4\) and \(n = 3\).

Step-by-step solution

Express one variable in terms of the other from the first equation

Isolate one of the variables, say, \(m\), in the first equation. The first equation is \(\frac{m}{2} + \frac{n}{3} - 3 = 0\). To isolate \(m\), we first move all terms not involving \(m\) to the other side: \(\frac{m}{2} = 3 - \frac{n}{3}\). Now, multiply both sides of the equation by 2 to solve for \(m\): \(m = 2(3 - \frac{n}{3}) = 6 - \frac{2n}{3}\).

Substitute the expression for one variable into the second equation

Now, we insert the expression for \(m\) found in Step 1 into the second equation. The second equation is \(\frac{n}{2} + \frac{m}{5} = \frac{23}{10}\). Substituting \(m\) we get: \(\frac{n}{2} + \frac{6 - \frac{2n}{3}}{5} = \frac{23}{10}\).

Solve the resulting equation for one variable

Combine the terms in the second equation and solve for \(n\). First, find a common denominator (which is 30) to combine the fractions: \(\frac{15n}{30} + \frac{6 \cdot 6 - 4n}{30} = \frac{69}{30}\). Simplify the left hand side: \(\frac{15n - 4n + 36}{30} = \frac{69}{30}\). Combine like terms: \(\frac{11n + 36}{30} = \frac{69}{30}\). Multiply both sides by 30 to get rid of the denominator: \(11n + 36 = 69\). Now subtract 36 from both sides: \(11n = 33\). Now divide by 11: \(n = 3\).

Substitute the found value into the expression from Step 1

Use the value of \(n=3\) and substitute it back into the expression for \(m\) derived in Step 1 to find \(m\). Thus, \(m = 6 - \frac{2n}{3} = 6 - \frac{2 \cdot 3}{3} = 6 - 2 = 4\).

Verification With Calculator

To check the solution, apply the values for \(m\) and \(n\) back into each original equation using a calculator to ensure both sides of the equation are equal. Since the verification process is done via calculator, provide the exact values to check, e.g., for the first equation input \(\frac{4}{2} + \frac{3}{3} - 3\) and for the second equation \(\frac{3}{2} + \frac{4}{5}\) and compare the results with each side of the equations.

Key concepts

System of Linear Equations

Understanding a system of linear equations is essential in mathematics and many other fields. It involves two or more linear equations that have common solutions. Think of each equation as a straight line on a graph. Where these lines intersect is where the solution lies — the point(s) that both (or all) equations have in common.

To solve a system of linear equations, there can be multiple methods such as graphing, substitution, elimination, and matrix operations. In the given exercise, we have a system of two equations with two variables, which typically has a single unique solution, as long as the equations are not parallel or the same line.

Substitution Method

The substitution method is like solving a puzzle by fitting pieces together one by one until you see the whole picture. You start by rearranging one equation to express one variable in terms of the other. For instance, if you have two variables, say 'x' and 'y', you solve one of the equations for 'x' and then replace 'x' with that expression in the other equation.

This method creates a single equation with just one variable, which is much more straightforward to solve. Once you have found the value of that variable, you substitute it back into one of the original equations to find the value of the other variable. This step-by-step process is a dependable way to work through a system of linear equations and reach a solution.

Algebraic Manipulation

Algebraic manipulation is the toolset that mathematicians use to transform equations. It involves various techniques like distributing, combining like terms, and moving terms across the equal sign by doing the opposite operation. For example, to clear fractions, you can multiply by a common denominator, and to isolate a term, you may add, subtract, multiply, or divide both sides of an equation by the same value.

In our exercise, algebraic manipulation comes into play when we multiply both sides to eliminate fractions, or when we combine like terms to simplify the equation. It's the art of rewriting equations to make them easier to understand and solve.

Equation Solving

Equation solving is akin to detective work, where one discovers the value of unknowns that make the equation true. It involves steps that simplify an equation or system of equations to find what values of the variables make the equation hold. This process requires a logical and structured approach, often employing other methods like substitution and algebraic manipulation.

The final aim is to arrive at a clear answer for each unknown in the equation. In the case of simultaneous equations, it's finding the exact numbers for our variables that satisfy all the equations at once. And like a good detective, one should always check that their found 'suspects' really fit by plugging them back into the original 'stories', or equations, to see if they truly work out.