Problem 86 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the system. Then check your solution. $$ \begin{array}{r} {2 m+3 n=7} \\ {m+n=1} \end{array} $$

Step-by-Step Solution

Answer

The solution to the system is \( m = -4 \) and \( n = 5 \)

Step 1: Multiply second equation

2(m+n) = 2, obtaining 2m + 2n = 2.

Step 2: Subtract second equation from the first

(2m + 3n) - (2m + 2n) = 7 - 2, simplifying to n = 5.

Step 3: Substitute n into second equation

m + 5 = 1, simplifying to m = -4.

Step 4: Check the solution

Substitute m = -4 and n = 5 into both original equations to confirm that they are true.

Key Concepts

Linear Equations

In algebra, linear equations are the simplest form of equations and serve as the foundation for understanding more complex mathematical concepts.
They are represented in the general form of \( ax + by = c \), where \( a \) and \( b \) are coefficients, \( x \) and \( y \) are variables, and \( c \) is a constant. The graph of a linear equation in two variables is a straight line in a two-dimensional plane.

Linear equations are critical because they model many real-world phenomena, such as calculating distances or predicting profits. When dealing with a system that includes two linear equations, the objective is to find the values of the variables that satisfy both equations simultaneously. In our example, the equations \(2m + 3n = 7\) and \(m + n = 1\) represent such a system, and by following a series of steps, we can solve for \(m\) and \(n\).

Substitution Method

The substitution method is a technique used to solve systems of equations. This method involves expressing one variable in terms of the other, and then substituting this expression into the other equation.
It is particularly useful when one equation in the system is easily solvable for one of the variables. Once a variable has been isolated, it can be replaced - or substituted - in the other equation, thereby reducing the system to a single equation with one variable.

For example, if we rearrange the second equation \(m + n = 1\) by isolating \(m\), we get \(m = 1 - n\). This expression for \(m\) can then be substituted into the first equation to find the value of \(n\). This step by step process ultimately leads to the solution of the system, which in our initial problem results in \(m = -4\) and \(n = 5\).

Systems of Equations

Systems of equations consist of two or more equations with the same set of variables. The goal when solving such systems is to find the values of the variables that satisfy all equations in the system at once.
A system of linear equations typically has a single solution (one point of intersection), infinitely many solutions (the lines coincide), or no solution (the lines are parallel).

In the context of our exercise, the system of equations given by \(2m + 3n = 7\) and \(m + n = 1\) can be solved using linear combinations, which involves adding or subtracting the equations in order to eliminate one of the variables. After finding the value for one variable, we can then find the value for the other variable. By strategically multiplying, adding, and subtracting the given equations, we can successfully determine the values of \(m\) and \(n\), which in this case are \(m = -4\) and \(n = 5\).