Problem 142 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 5 x-3 y=-1 \\ 2 x-y=2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((x, y) = (7, 12)\).

Step 1: Solve one equation for one variable

Start with the second equation, since it is simpler to isolate a variable. Solve for y: \(2x - y = 2\) Add \(y\) to both sides: \(2x = y + 2\) Now subtract 2 from both sides: \(y = 2x - 2\)

Step 2: Substitute into the other equation

Substitute \(y = 2x - 2\) into the first equation: \(5x - 3y = -1\) Replace \(y\) with \(2x - 2\): \(5x - 3(2x - 2) = -1\)

Step 3: Simplify and solve for x

Distribute the \(3\) across \((2x - 2)\): \(5x - 6x + 6 = -1\) Simplify the left side: \(-x + 6 = -1\) Subtract 6 from both sides: \(-x = -7\) Multiply both sides by \(-1\): \(x = 7\)

Step 4: Solve for the second variable

Substitute \(x =7\) back into \(y = 2x - 2\): \(y = 2(7) - 2\) Calculate the value: \(y = 14 - 2\) \(y = 12\)

Step 5: Write the solution

The solution to the system of equations is: \((x, y) = (7, 12)\)

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving systems of linear equations. In this method, we isolate one variable in one of the equations and then substitute its expression into the other equation. This helps eliminate one variable and allows us to solve for the other.

For example, in the given problem, we start by solving the second equation for y:
\(2x - y = 2\).
Adding y to both sides, we get:
\(2x = y + 2\).
Next, subtract 2 from both sides to isolate y:
\(y = 2x - 2\).
This equation tells us that wherever we see y in the first equation, we can replace it with \(2x - 2\).

Once y is substituted, the system becomes a single-variable equation which is much easier to solve. This method ensures that all possible solutions are explored.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true.

After substituting the expression for y into the first equation, we get:
\(5x - 3(2x - 2) = -1\).
Distributing \(-3\) across \(2x - 2\), we obtain:
\(5x - 6x + 6 = -1\).
Now, simplify the equation:
\(-x + 6 = -1\).
To isolate x, first subtract 6 from both sides:
\(-x = -7\).
Finally, multiply both sides by \(-1\):
\(x = 7\).

With x determined, we can use its value to find y by plugging it back into the equation \(y = 2x - 2\). Thus, \(y = 2(7) - 2 = 14 - 2 = 12\).
This gives us the solution to the system:
\((x, y) = (7, 12)\).

Algebraic Manipulation

Algebraic manipulation is essential for solving equations. It involves techniques such as addition, subtraction, multiplication, division, and distribution to simplify and solve equations.

Let's look at the operation in our problem:

• Addition and subtraction to move terms from one side of the equation to the other.
• Multiplication to simplify the expression by distributing coefficients.
• Division to isolate the variable.

For instance, in solving \(-x + 6 = -1\), we used subtraction to remove the constant term, then multiplication to isolate the variable x. These steps show how manipulating equations algebraically is vital for finding solutions efficiently.

Mastering these algebraic techniques simplifies even complex problems, making the process of solving systems of equations straightforward and manageable.