Problem 76 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} -2 x+2 y=6 \\ y=-3 x+1 \end{array}\right. $$

Step-by-Step Solution

Answer

(x, y) = (-0.5, 2.5)

Step 1: Substitute the value of y

The second equation is already solved for y. We have: y = -3x + 1Now substitute yyy from this equation into the first equation.

Step 2: Solve for x

Substitute the value of y into the first equation: -2x + 2(-3x + 1) = 6Simplify and solve for x:-2x - 6x + 2 = 6-8x + 2 = 6-8x = 4x = -0.5

Step 3: Solve for y using the value of x

Substitute the value of x back into the second equation to find y:y = -3x + 1y = -3(-0.5) + 1y = 1.5 + 1y = 2.5

Key Concepts

substitution method

The substitution method is a useful technique for solving systems of equations. It involves isolating one variable in one of the equations and then substituting that expression into the other equation. This simplifies the system and allows us to solve for each variable step by step.

In this exercise, we start with the system:
\[ \begin{cases} -2x + 2y = 6 \ y = -3x + 1 \ \end{cases} \]
Notice that the second equation is already solved for \( y \). This is perfect for the substitution method because we can directly use this expression to replace \( y \) in the first equation.

\( y = -3x + 1 \)
We substitute this into the first equation:
\[ -2x + 2(-3x + 1) = 6 \]

This step reduces the two-variable system to a single equation with one variable, making it much easier to solve.

solving linear equations

Once we substitute \( y \) with \( -3x + 1 \), our goal is to solve the resulting linear equation for \( x \). We continue from:
\[ -2x + 2(-3x + 1) = 6 \]

We first distribute the 2 inside the parenthesis:
\[ -2x - 6x + 2 = 6 \]
Next, combine the \( x \) terms:
\[ -8x + 2 = 6 \]
Subtract 2 from both sides:
\[ -8x = 4 \]
Then, divide by -8 to solve for \( x \):
\[ x = -0.5 \]

This method follows core algebraic principles, such as distributing, combining like terms, and isolating the variable, ensuring that each step is logically sound and that we maintain equality throughout.

algebraic manipulation

Algebraic manipulation is imperative when dealing with systems of equations and involves operations like addition, subtraction, multiplication, and division to simplify equations.

Here, we've already solved for \( x \). Now, we need to find the value of \( y \). Using our solved value \( x = -0.5 \), we substitute it back into the second equation:
\( y = -3(-0.5) + 1 \)
First, handle the multiplication:
\[ y = 1.5 + 1 \]
Then, add the constants:
\[ y = 2.5 \]

Through careful algebraic manipulation, we determined that when \( x = -0.5 \), the corresponding value of \( y \) is 2.5. Each algebraic step is crucial for ensuring accuracy and comprehensibility in solving systems of equations.