Problem 13 - College Algebra Essentials

Question

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 2 x+5 y=-4 \\ 3 x-y=11 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is x = 3, y = -2

Step 1: Isolate one variable in one equation

Isolate the variable y in equation 2:\n3x - y = 11 implies y = 3x - 11

Step 2: Substitute the value of y into the other equation

Substitute y = 3x-11 from equation 2 into equation 1: \n2x + 5(3x - 11) = -4, which simplifies to 17x = 51

Step 3: Solve for variable x

Solve the equation from the second step to find x: \n17x = 51 gives x = 3

Step 4: Substitute value of x into y equation

Substitute x = 3 into the previously isolated y equation: \ny = 3(3) - 11 = -2

Step 5: Check the solution

Substitute x = 3 and y = -2 in the original equations:\n2(3) + 5(-2) = -4 and 3(3) - (-2) = 11. Both are true, therefore the solution is correct

Key Concepts

Understanding Systems of Equations

A system of equations is a set of two or more equations that have common variables. These equations work together, and we're tasked with finding values for these variables that satisfy all the equations simultaneously. In our example, the equations provided are:

• \(2x + 5y = -4\)
• \(3x - y = 11\)

Both of these equations contain two variables, \(x\) and \(y\). The goal is to find a pair \((x, y)\) that makes both equations true. There are different methods to solve systems of equations, like substitution or elimination. Here, we are using the substitution method. This method involves isolating one variable and substituting its expression into the other equation, which eventually helps us to find specific solutions for the variables. This is crucial in solving algebra problems as it provides a systematic approach to finding solutions.

Linear Equations in Algebra

Linear equations are a type of equation where the variables appear in a linear form, meaning they have no exponents or powers other than one. They are represented graphically as straight lines in a coordinate plane.

• In the system \(2x + 5y = -4\) and \(3x - y = 11\), both equations are linear.
• These equations can be expressed in the form \(Ax + By = C\) which generally describes a line with slope and y-intercept.

Understanding that these are linear equations is essential because it tells us that the solutions will correspond to points where these lines intersect. This is important in algebra problem solving as it simplifies the process, knowing the behavior and representation of the equation provides a map to the solution.

Effective Algebra Problem Solving Using Substitution Method

The substitution method is an effective technique for solving a system of equations, especially when one equation can be easily solved for one of the variables. Let's delve deeper into how this method is applied.
By isolating \(y\) in the equation \(3x - y = 11\), we obtained \(y = 3x - 11\). This expression for \(y\) is then substituted into the other equation.
Here are the steps broken down:

• Isolate one variable: Choose an equation, like \(3x - y = 11\), and express \(y\) in terms of \(x\).
• Substitution: Replace \(y\) in the first equation \(2x + 5y = -4\) to have a single-variable equation \(2x + 5(3x - 11) = -4\).
• Solve for the first variable: Simplify and solve for \(x\), giving \(17x = 51\) which simplifies to \(x = 3\).
• Solve for the second variable: Substitute \(x = 3\) back into \(y = 3x - 11\) to find \(y = -2\).
• Verify: Ensure both values satisfy the original equations, safeguarding accurate solutions.

The substitution method, if mastered, aids in solving complex problems efficiently and is a pivotal skill for tackling algebra problems.