Problem 28 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} -x+3 y=18 \\ -3 x+2 y=19 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -3\) and \(y = 5\).

Step 1: Solve First Equation for x

From the first equation, we have \[-x + 3y = 18.\]We can rearrange it to express \(x\) in terms of \(y\):\[-x = 18 - 3y \Rightarrow x = 3y - 18.\]

Step 2: Substitute Expression into Second Equation

Now substitute \(x = 3y - 18\) into the second equation:\[-3(3y - 18) + 2y = 19.\]Simplify and solve for \(y\):\[-9y + 54 + 2y = 19 \rightarrow -7y + 54 = 19.\]

Step 3: Solve for y

Subtract 54 from both sides of the equation:\[-7y = 19 - 54 \Rightarrow -7y = -35.\]Then divide both sides by -7:\[y = 5.\]

Step 4: Substitute y back to find x

Use the value of \(y\) in the expression for \(x\):\[x = 3(5) - 18 = 15 - 18 = -3.\]

Step 5: Conclusion: Solution to the System

The solution to the system of equations is:\[x = -3, \, y = 5.\]

Key Concepts

Understanding a System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In our original exercise, the system consists of two equations with variables \(x\) and \(y\):

• \(-x + 3y = 18\)
• \(-3x + 2y = 19\)

The goal in solving a system of equations is to find the set of values for the variables that satisfy all given equations simultaneously.
This means the solution should make all equations in the system true when substituted into them. Using methods like substitution or elimination, we can solve these systems effectively. The substitution method, in particular, is useful when one of the equations can be easily rearranged to express one variable in terms of the other.

Mastering the Art of Solving Equations

Solving equations is a crucial skill in algebra that involves finding the value of the variable that makes the equation true. In the substitution method, we're leveraging this skill by substituting the expression of one variable into another equation.### Steps to Solve Using Substitution Method - **Rearrange** one of the equations to express one variable in terms of the other. For instance, from the first equation in the task, solve for \(x\): \[-x + 3y = 18 \Rightarrow x = 3y - 18\]- **Substitute** this expression in the other equation. Replace \(x\) with \(3y - 18\) in the second equation: \[-3(3y - 18) + 2y = 19\]- **Solve** for the remaining variable. Here, solving for \(y\) gives us its value, \(y = 5\).- **Back-substitute** the found value to get the other variable. Use \(y = 5\) to find \(x\): \[x = 3(5) - 18 = -3\]Each step is purposefully designed to simplify calculation and help derive the correct solution.

Breaking Down the Algebra Steps

Algebra involves a series of steps to simplify and solve complex equations. Let's detail the key algebra steps in the substitution method:**Step 1: Rearrange**- Identify which equation can be easily manipulated. Rearrange it to express one variable in terms of another, as we did with \(x = 3y - 18\). **Step 2: Substitute and Simplify**- Carefully substitute the expression into the other equation. Simplifying these subtitutions involves multiplying, adding, or subtracting terms.**Step 3: Solve for One Variable**- Once simplified, solve for one of the variables. It's crucial to perform operations symmetrically on both sides, such as subtracting or adding terms.**Step 4: Back-substitute**- After finding the value of one variable, replace it back into any initial expressions to find the second variable.This process requires precision at each step to avoid mistakes and ensure an accurate solution. It's the consistency in applying these steps that reinforces the overall understanding of solving systems using substitution.