Problem 79 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

x = 3, y = -1

Step 1: Solve the second equation for y

The second equation is already solved for y: \[ y = \frac{1}{3}x - 2 \].

Step 2: Substitute y in the first equation

Substitute \( y = \frac{1}{3}x - 2 \) into the first equation: \[ 2x + 5 \left( \frac{1}{3}x - 2 \right) = 1 \].

Step 3: Simplify the equation

Distribute the 5: \[ 2x + \frac{5}{3}x - 10 = 1 \].Combine the terms with x: \[ \left( 2 + \frac{5}{3} \right)x - 10 = 1 \].

Step 4: Find a common denominator

Convert 2 to an equivalent fraction: \[ \frac{6}{3} \].Now the equation is: \[ \frac{6}{3}x + \frac{5}{3}x - 10 = 1 \].Combine the fractions: \[ \frac{11}{3}x - 10 = 1 \].

Step 5: Solve for x

Isolate x by adding 10 to both sides: \[ \frac{11}{3}x = 11 \].Multiply both sides by 3/11: \[ x = 3 \].

Step 6: Find y

Substitute \( x = 3 \) back into \( y = \frac{1}{3}x - 2 \): \[ y = \frac{1}{3}(3) - 2 \].Calculate y: \[ y = 1 - 2 = -1 \].

Key Concepts

substitution method

The substitution method is a powerful technique for solving systems of equations. It involves isolating one variable in one of the equations and substituting this expression into the other equation. This transforms the system into a single equation with one variable, making it much easier to handle.

solving linear equations

Solving linear equations is a fundamental skill in algebra. It typically involves isolating the variable (often denoted as x or y) by performing operations such as addition, subtraction, multiplication, and division. The goal is to find the value of the unknown that makes the equation true.

algebraic manipulation

Algebraic manipulation includes various techniques used to simplify and solve equations. These techniques range from distributing multiplication over addition to combining like terms and factoring. Algebraic manipulation is crucial for transforming complex expressions into simpler, more manageable forms.

common denominators

Finding common denominators is essential when dealing with fractions in equations. This process involves converting fractions to equivalent fractions with the same denominator, which allows for easy addition or subtraction of the fractions. For example, converting 2 to \[ \frac{6}{3} \] ensures it can be combined seamlessly with \[ \frac{5}{3} \].