Problem 25 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} 3 x-y=1 \\ 2 x-3 y=10 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = -1 \), \( y = -4 \).

Step 1: Solve the first equation for y

Start by isolating the variable \( y \) in the first equation. From \( 3x - y = 1 \), add \( y \) to both sides and then subtract 1 from both sides to get:\[y = 3x - 1\]

Step 2: Substitute y in the second equation

Take the expression for \( y \) from Step 1 and substitute it into the second equation, \( 2x - 3y = 10 \). Replace \( y \) with \( 3x - 1 \):\[2x - 3(3x - 1) = 10\]

Step 3: Simplify the substituted equation

Distribute \(-3\) into the expression \( 3x - 1 \) in the equation:\[2x - 9x + 3 = 10\]Now, combine like terms:\[-7x + 3 = 10\]

Step 4: Solve for x

Subtract 3 from both sides of the equation:\[-7x = 7\]Now, divide both sides by \(-7\) to solve for \( x \):\[x = -1\]

Step 5: Substitute x back into the expression for y

With the value of \( x = -1 \), substitute it back into the expression \( y = 3x - 1 \):\[y = 3(-1) - 1\]Simplify to find \( y \):\[y = -3 - 1 = -4\]

Step 6: Verify the solution

Substitute \( x = -1 \) and \( y = -4 \) back into the original equations to ensure they satisfy both equations:1. For \( 3x - y = 1 \): \( 3(-1) - (-4) = -3 + 4 = 1 \) (True)2. For \( 2x - 3y = 10 \): \( 2(-1) - 3(-4) = -2 + 12 = 10 \) (True)Since both are true, the solution is verified.

Key Concepts

Substitution Method

The substitution method is a powerful technique in solving systems of equations where you solve one equation for one variable and then substitute that expression into the other equations. In simpler terms, it's like finding a code in one part of the puzzle and using it to unlock another part. This process is particularly useful for solving systems of two linear equations as it directly isolates one variable, making calculations more straightforward.
For example, consider the system:

• \( 3x - y = 1 \)
• \( 2x - 3y = 10 \)

We first solve the first equation for \( y \), which gives us \( y = 3x - 1 \). Notice how we've expressed \( y \) in terms of \( x \). This expression is then substituted into the second equation, replacing \( y \) with \( 3x - 1 \).
This substitution transforms the problem into a single-variable equation that can be solved more easily. Once we find the value of \( x \), we use it to find \( y \) by plugging it back into the expression \( y = 3x - 1 \). The substitution method simplifies the otherwise daunting task of juggling two equations into a series of more manageable steps.

Linear Equations

Linear equations represent relationships between variables with predictable, straight-line behavior on a graph. They are often written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
In our example, we have:

• \( 3x - y = 1 \)
• \( 2x - 3y = 10 \)

These equations are linear because they involve variables raised only to the first power. They imply a straight-line relation when plotted on a graph. In solving such a system, our goal is to find a pair of \( x \) and \( y \) values that satisfies both equations simultaneously. Depending on the equations, there can be a single solution (where the lines intersect), no solution (if the lines are parallel), or infinitely many solutions (if the lines coincide).
The substitution method shows that these variables are dependent on each other, so by finding \( x \), you can determine \( y \), demonstrating the interconnectedness of the equations.

Step-by-Step Solutions

Solutions to a system of linear equations require a structured approach to ensure accuracy and understanding. The step-by-step method allows students to visualize the process and understand each part of the solution individually.
Let's walk through the process we used:

• **Step 1**: Solve one equation for one variable. We chose the first equation and solved for \( y \): \( y = 3x - 1 \).
• **Step 2**: Substitute this expression into the other equation. We replaced \( y \) in the second equation with \( 3x - 1 \).
• **Step 3**: Simplify and solve for the remaining variable, \( x \). We found \( x = -1 \).
• **Step 4**: Use the known \( x \) to find \( y \) by substituting back: \( y = -4 \).
• **Step 5**: Verify the solution by substituting both values into the original equations to ensure both equations hold true.

Step-by-step solutions not only make the solving process methodical but also increase confidence that the solution is correct. Breaking down tasks helps in learning and facilitates tackling more complex equations in the future.