Problem 91 - College Algebra with Modeling and Visualization

Question

Solve the system, if possible. $$ \begin{aligned} 7 x-5 y &=-15 \\ -2 x+3 y &=-2 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \(x = -5, y = -4\).

Step 1: Identify the System of Equations

We need to solve the system of linear equations: \( 7x - 5y = -15 \) and \( -2x + 3y = -2 \). Our goal is to find values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 1: Eliminate x

To eliminate \(x\), we first need a common coefficient for \(x\) in both equations. Multiply the first equation by 2 and the second by 7 to achieve this:\[2(7x - 5y) = 2(-15) \quad \Rightarrow \quad 14x - 10y = -30\]\[7(-2x + 3y) = 7(-2) \quad \Rightarrow \quad -14x + 21y = -14\]

Step 2: Add the Equations

Add the newly obtained equations:\[(14x - 10y) + (-14x + 21y) = -30 + (-14)\]The \(x\) terms cancel out, leaving:\[11y = -44\]

Step 3: Solve for y

Divide both sides of the equation by 11 to solve for \(y\):\[y = \frac{-44}{11} = -4\]

Step 4: Substitute y back into an Equation

Substitute \(y = -4\) back into one of the original equations, say \(7x - 5y = -15\):\[7x - 5(-4) = -15 \quad \Rightarrow \quad 7x + 20 = -15\]

Step 5: Solve for x

Subtract 20 from both sides:\[7x = -15 - 20 \quad \Rightarrow \quad 7x = -35\]Divide by 7:\[x = \frac{-35}{7} = -5\]

Step 6: Verify the Solution

Substitute \(x = -5\) and \(y = -4\) into the second equation \(-2x + 3y = -2\) to verify:\[-2(-5) + 3(-4) = 10 - 12 = -2\]Both equations are satisfied, confirming the solution is correct.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical statements involving constants and variables that can be represented as straight lines when plotted on a graph. In a linear equation, each term is either a constant or the product of a constant and a single variable.

Consider the equation in the form of:

• \( ax + by = c \)

Here, \(a\), \(b\), and \(c\) are constants, while \(x\) and \(y\) are variables.

The goal when solving linear equations is to find the value of the variables that make the equation true.

In a system of linear equations, we deal with more than one equation. A solution is a set of values that satisfy all the equations in the system simultaneously.

Exploring the Substitution Method

The substitution method is a fundamental technique for solving systems of linear equations. It works by expressing one variable in terms of the other using one equation and then substituting this expression into another equation.

Steps to apply the substitution method:

• Solve one of the equations for one variable in terms of the other.
• Substitute this expression into the second equation to find the value of the other variable.
• Use the found value to solve for the initial variable.

This method is particularly useful when one of the equations can be easily manipulated to express one variable in terms of the other.

This process simplifies the system of equations, making it easier to find a precise solution.

Understanding the Elimination Method

The elimination method, also known as the addition method, is a strategic way to solve systems of equations by eliminating variables. By adding or subtracting entire equations, the coefficients of one variable can be canceled, making it possible to solve for the other variable.

Steps to use the elimination method:

• Align the equations so that like terms and variables are arranged.
• Multiply each equation by a necessary coefficient to align the coefficients of one variable.
• Add or subtract the equations to eliminate one variable.
• Solve for the remaining variable, then substitute back to find the other variable.

The elimination method is ideal when the equations can be easily manipulated to cancel out one of the variables, offering a clear path to the solution.

Solving Systems of Equations

Systems of equations involve solving multiple equations simultaneously. Each equation represents a linear relationship, and the goal is to find a common solution that satisfies all equations at once.

In the given problem:

• We have a system of two equations: \( 7x - 5y = -15 \) and \( -2x + 3y = -2 \).

There are two popular methods to solve such systems:

• Substitution Method
• Elimination Method

Finding where these linear equations intersect through either method yields the solution. For this problem, the solution is \( x = -5 \) and \( y = -4 \), which satisfies both equations when substituted back, confirming the accuracy of the solution.

Systems of equations are pivotal in mathematics as they model numerous real-world scenarios where multiple conditions or constraints exist.