Problem 38 - Algebra 2

Question

Solve each system by elimination. \(\left\\{\begin{aligned} r+3 s &=7 \\ 2 r-s &=7 \end{aligned}\right.\)

Step-by-Step Solution

Answer

The solution to the system is: \(r = 4, s = 1\).

Step 1: Multiply second equation for elimination

To eliminate one variable, multiply the second equation by 3 to make the coefficients of the variable 's' opposite in each equation.

Step 2: Create the new system after multiplication

After multiplying the second equation by 3, the system becomes: \[\left\{\begin{aligned} r+3s &= 7 \ 6r-3s &= 21 \end{aligned}\right.\] Now, the 's' terms have coefficients that are additive inverses of each other.

Step 3: Add the equations to eliminate 's'

Add the two equations to eliminate variable 's': \((r + 3s) + (6r - 3s) = 7 + 21\).

Step 4: Solve for 'r'

After adding the equations, we get \(7r = 28\). Divide both sides by 7 to solve for 'r': \(r = \frac{28}{7}\).

Step 5: Solve for 's' using first equation

Substitute the value of 'r' back into the first equation to solve for 's': \(r + 3s = 7\) substitute the value of 'r', we get \(4 + 3s = 7\).

Step 6: Isolate and solve for 's'

Subtract 4 from both sides to isolate 's': \(3s = 7 - 4\). Then divide by 3: \(s = \frac{3}{3}\).

Key Concepts

Elimination Method

The elimination method is a standard technique in algebra for solving systems of linear equations. Its core principle involves combining equations to eliminate one variable, making it easier to solve for the other. This process typically involves either adding or subtracting the equations after manipulating them so that the coefficients of one variable are opposites.
To apply this method effectively, you sometimes need to multiply one or both equations by a number to create a scenario where adding or subtracting the equations will eliminate one variable. Once a variable is eliminated, you have a simpler equation with just one variable to solve.
After finding the value of the first variable, substitute it back into one of the original equations to solve for the second variable. The elimination method is particularly handy when dealing with large systems of equations or when the coefficients of the variables line up neatly for elimination.

Algebraic Equations

An algebraic equation is a mathematical statement that indicates that two algebraic expressions are equal. It contains one or more variables and is solved by finding the value of the variables that make the equation true. In the context of a system of equations, you'll have a set of two or more equations with the same set of variables.
These equations represent relationships between quantities, often allowing us to solve real-world problems. The process of solving algebraic equations involves various techniques, such as simplifying expressions, combining like terms, and manipulating the equations to isolate variables. Understanding how to manipulate these equations is fundamental to the study of algebra and is essential for solving more complex problems in mathematics and related fields.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The goal when solving such a system is to find the values of the variables that satisfy all equations in the system simultaneously. Solutions to the system can be interpreted graphically as the point or points where the lines represented by the equations intersect.
There are three possible outcomes when solving a system of linear equations: one solution (the lines intersect at one point), infinitely many solutions (the lines are coincident), or no solution (the lines are parallel and never intersect).
Solving these systems can be accomplished through various methods, with the elimination method being one among others like substitution and graphical methods. The choice of method often depends on the structure of the equations and personal preference for the convenience of calculations.