Problem 15 - College Algebra Essentials

Question

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 2 x-3 y=8-2 x \\ 3 x+4 y=x+3 y+14 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 17\) and \(y = 20\).

Step 1: Simplify Equations

Simplify the given equations as follows,1) From \(2x - 3y = 8 - 2x\), simplify it to \(4x - 3y = 8\).2) From \(3x + 4y = x + 3y + 14\), simplify it to \(2x - y = 14\).

Step 2: Solve for One Variable

Take the second equation and solve for one variable. Here, let's solve for y,\(2x - y = 14\) becomes \(y = 2x - 14\).

Step 3: Substitute into First Equation

Substitute \(y = 2x - 14\) into the first simplified equation to solve for x,\(4x - 3(2x - 14) = 8\),\(4x - 6x + 42 = 8\),\(-2x + 42 = 8\),\(-2x = -34\),\(x = 17\).

Step 4: Solve for Other Variable

Substitute x = 17 into the equation solved for y from Step 2,\(y = 2(17) - 14\),\(y = 34 - 14\),\(y = 20\).

Key Concepts

System of Equations

When dealing with a system of equations, we are working with two or more equations that share the same set of unknowns. In this particular exercise, we have a pair of equations featuring the variables \(x\) and \(y\), which are intertwined. Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

• The essence of a system of equations is to determine the common solution that all equations agree upon.
• These systems can be solved through various methods, such as substitution, elimination, or graphing.
• The substitution method involves solving one equation for a single variable and then substituting that expression into another equation.

In this exercise, the substitution method is chosen, which necessitates isolating one variable in one equation to insert it into another equation, demonstrating the interplay and interchangeability between equations in a system. This method is particularly useful for linear equations as seen here.

Algebraic Simplification

Simplifying equations is a pivotal step in solving them efficiently. The goal is to rewrite the equations in a form that is easier to manipulate.

• In the given exercise, both initial equations undergo a simplification process to make them more manageable.
• For instance, the equation \(2x - 3y = 8 - 2x\) is simplified by moving terms involving \(x\) to one side resulting in \(4x - 3y = 8\).
• Similarly, \(3x + 4y = x + 3y + 14\) simplifies to \(2x - y = 14\) after rearranging terms.

This simplification helps us clearly see the relationships between the variables and makes finding a solution via substitution or other methods easier. Algebraic simplification reduces potential calculation errors and clarifies the path to the solution.

Solving for Variables

Solving for variables involves isolating a variable to determine its value. In this exercise, once the equations are simplified, each variable is solved step by step.

• Initially, from the second simplified equation \(2x - y = 14\), we solve for \(y\) by expressing it in terms of \(x\). This gives \(y = 2x - 14\).
• This expression is then substituted back into the first equation \(4x - 3y = 8\) to solve for \(x\).
• After substituting, the calculations simplify to \(-2x = -34\), leading to \(x = 17\).

Finally, with \(x = 17\), we substitute back into \(y = 2x - 14\) to find \(y = 20\). This process effectively solves the system of equations, pinpointing the exact values of \(x\) and \(y\) where both equations are satisfied. Each variable's isolation and calculation underscore the reliability of systematic, stepwise solving methods.