Problem 31 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{rl}x-3 y & =-22 \\ 2 x+7 y & =60\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \( x = 2 \) and \( y = 8 \).

Step 1: Solve the First Equation for x

Take the first equation: \( x - 3y = -22 \). Rearrange it to solve for \( x \):\[ x = 3y - 22\]

Step 2: Substitute x in the Second Equation

Substitute the expression for \( x \) from Step 1 into the second equation: \( 2x + 7y = 60 \).Replace \( x \) with \( 3y - 22 \):\[2(3y - 22) + 7y = 60\]

Step 3: Simplify and Solve for y

Expand the equation:\[6y - 44 + 7y = 60\]Combine like terms:\[13y - 44 = 60\]Add 44 to both sides:\[13y = 104\]Divide by 13:\[y = 8\]

Step 4: Substitute y Back to Find x

Use the value of \( y \) (8) found in Step 3 and substitute it back into the expression for \( x \) from Step 1:\[x = 3(8) - 22\]Calculate \( x \):\[x = 24 - 22 = 2\]

Step 5: State the Solution

The solution to the system of equations is \( x = 2 \) and \( y = 8 \).

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations that share the same set of variables. In this exercise, we are dealing with a system made up of two linear equations. Solving a system of equations means finding the values of the variables that satisfy all equations in the system simultaneously.
The two equations given in the form are:

• Equation 1: \(x - 3y = -22\)
• Equation 2: \(2x + 7y = 60\)

One way to solve such systems is by using the substitution method, which involves expressing one variable in terms of another using one equation and then substituting this expression into the remaining equations. This reduces the number of equations and makes it easier to find the solution without having to guess or check multiple possibilities.

Linear Equations

Linear equations are equations of the first degree, which means they involve only the powers of one for the variables. Each term in a linear equation is either a constant or a product of a constant and a single variable. In their graphical representation, linear equations form straight lines.
Here are some basic properties of linear equations:

• They can be written in the standard form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.
• The solution to a linear equation gives the value of the variable(s) that make the equation true.

In our system, both equations are linear, making it possible to apply algebraic methods like substitution to find an exact solution. When combined in a system, these equations represent lines in a coordinate plane, and the solution is the point where these lines intersect.

Algebraic Manipulation

Algebraic manipulation is the technique of rearranging and simplifying algebraic expressions to solve for unknown variables. This often involves operations like addition, subtraction, multiplication, and division to isolate a variable.
For our problem, the steps involve several key algebraic manipulations:

• Isolate a variable: In Step 1, we solve for \(x\) from the first equation to get \(x = 3y - 22\). This is a crucial step in the substitution method.
• Substitute: The expression for \(x\) is substituted into the second equation, effectively reducing the system to a single equation with one variable, \(y\).
• Simplify and Solve: In Steps 3 and 4, we combine like terms and perform arithmetic operations to solve for the variables \(y\) and \(x\).

Through these manipulations, the problem is broken down into simpler parts, allowing for step-by-step resolution and consistent logic application.