Problem 20 - Algebra 2

Question

Solve each system by elimination. \(\left\\{\begin{aligned} 3 a+4 b &=9 \\\\-3 a-2 b &=-3 \end{aligned}\right.\)

Step-by-Step Solution

Answer

By elimination, the solution to the system is a = 1, b = 1.5.

Step 1: Add Both Equations

Since one equation has a positive coefficient for a and the other equation has a negative coefficient for a (and they are equal in magnitude), add the two equations directly to eliminate the variable a.

Step 2: Solve for b

After adding the equations, solve the resulting equation for b. Combine like terms and isolate b on one side of the equation.

Step 3: Substitute b's Value into One of the Original Equations

Take the value found for b and substitute it back into one of the original equations to find the value of a.

Step 4: Solve for a

With b substituted in, solve for a, isolating a on one side to find its value.

Key Concepts

Elimination Method Algebra

The elimination method, a cornerstone of algebra, is a technique used for solving systems of linear equations; it involves combining the equations in a way that cancels out one of the variables, allowing for the solution of the remaining variable.

In practical terms, students can employ this method whenever they encounter pairs of equations where the coefficients of one variable are opposites or can be made opposite through multiplication. Once the variable is eliminated, the remaining equation with a single variable is significantly easier to solve.

It's crucial in this method to line up the equations, add or subtract them carefully, and avoid simple algebraic errors. This ensures a successful elimination of one variable and paves the way to find the solution of the equations.

Simultaneous Equations

Simultaneous equations are sets of equations with multiple variables where all the equations are true at the same time. It's like finding the meet-up point of different paths that each variable's value is traveling along.

To solve simultaneous equations, we manipulate the equations until we isolate a single variable. This often involves the elimination method, substitution, or graphing. They are referred to as 'simultaneous' because the variables need to work together, simultaneously, to satisfy all equations in the system.

Importance in Algebra

Solving these kinds of equations is fundamental in algebra and is foundational for understanding more complex mathematics, such as in calculus and linear algebra. They are also practical in real-life situations where multiple conditions need to be met simultaneously.

Solving Linear Systems

Solving linear systems is the process of finding the values of the variables that satisfy all linear equations in the system at once. These systems can be represented as lines on a graph and the solutions are seen where these lines intersect.

Such systems can have one unique solution, no solution (if the lines are parallel and never intersect), or infinitely many solutions (if the lines are coincident, meaning they lie one on top of the other). The method of elimination is particularly effective for finding the unique solution when it exists, by reducing the system to a simpler one that is easier to examine and resolve.

Understanding how to work with linear systems is not only crucial for academic progression in mathematics but is also applicable in various scientific fields, including engineering, economics, and social sciences, where relationships between quantities often need to be examined and resolved.