Problem 33 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{l}4 x-5 y=21 \\ 3 x+7 y=-38\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \(x = -1, y = -5\).

Step 1: Solve the First Equation for x

Start with the first equation, \(4x - 5y = 21\). To use substitution, first solve this equation for \(x\). Add \(5y\) to both sides to get \(4x = 5y + 21\). Then, divide every term by 4: \(x = \frac{5y + 21}{4}\).

Step 2: Substitute x in the Second Equation

Now, substitute \(x = \frac{5y + 21}{4}\) into the second equation, \(3x + 7y = -38\). Substitute to get \(3\left(\frac{5y + 21}{4}\right) + 7y = -38\). Multiply through to eliminate the fraction: \(\frac{15y + 63}{4} + 7y = -38\).

Step 3: Clear the Fraction

Multiply every term by 4 to clear the fraction in \(\frac{15y + 63}{4} + 7y = -38\). This gives \(15y + 63 + 28y = -152\).

Step 4: Combine Like Terms

Combine \(15y\) and \(28y\) to simplify the equation: \(43y + 63 = -152\).

Step 5: Solve for y

Subtract 63 from both sides: \(43y = -215\). Then, divide by 43: \(y = -5\).

Step 6: Substitute y Back into Expression for x

Use \(y = -5\) in the expression for \(x\): \(x = \frac{5(-5) + 21}{4}\). Simplify to find \(x = \frac{-25 + 21}{4} = \frac{-4}{4} = -1\).

Step 7: State the Solution

The solution to the system of equations is \((x, y) = (-1, -5)\).

Key Concepts

Understanding Systems of Equations

A system of equations consists of two or more equations with common variables. In the presented exercise, we have two equations featuring variables \(x\) and \(y\):

• \(4x - 5y = 21\)
• \(3x + 7y = -38\)

Systems of equations can describe real-world situations where multiple conditions occur simultaneously. In algebra, solving a system means finding the values of the variables that satisfy all equations at the same time.

These solutions are often represented as a point \((x, y)\) in a coordinate plane where the graphs of the equations intersect. Understanding how to solve systems like these is essential, as it applies to various fields such as physics, engineering, and economics.

Steps to Solving Equations

Solving equations is a foundational skill in mathematics that involves finding the value of variables that make the equation true. The chosen method for this exercise is substitution, which is particularly useful when one equation is easily solvable for one variable.

Here’s a step-by-step guide:

• **Initial Isolation**: Start by isolating one variable in one of the equations. In our example, the first equation was rearranged to solve for \(x\): \(x = \frac{5y + 21}{4}\).
• **Substitution**: Substitute this expression into the other equation. This allows you to work with a single variable, making it simpler to solve.
• **Simplification**: Simplify the resulting equation, combining like terms and clearing fractions if needed, as seen by multiplying every term by 4.
• **Solution**: Solve for the isolated variable. In this case, \(y\) was determined first as \(y = -5\).
• **Back-substitution**: Use the found value to determine the other variable. Here, substituting \(y = -5\) found \(x = -1\).

Solving equations with substitution requires precise manipulation and clear calculations to avoid errors, making attention to detail crucial.

Algebra Fundamentals

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's essential for understanding patterns, relationships, and changes in various situations. In this exercise, algebra is used to simplify and solve equations systematically.

Key Algebra Concepts:

• **Variables**: Symbols, often letters, used to represent unknown values in equations. Here, \(x\) and \(y\) are variables.
• **Expression Manipulation**: Rearrange and simplify equations to isolate variables using mathematical operations like addition, subtraction, multiplication, and division.
• **Equality Property**: Ensuring both sides of an equation remain equal by performing the same operation, crucial in maintaining balance in an equation.

Algebra serves as the language of mathematics for more complex topics, forming the basis for calculations in calculus, statistics, and beyond.

Exploring Linear Equations

Linear equations are equations of the first degree, meaning they contain no exponents higher than one. They form straight lines when graphed on a coordinate plane.

The equations in our system are both linear:

• \(4x - 5y = 21\)
• \(3x + 7y = -38\)

Characteristics of Linear Equations include:

• **Straight Lines**: Each equation plots a line, characterized by its slope and intercept, which is the point where it crosses the axis.
• **Relationships**: The intersection of these lines represents the solution to a system of linear equations, indicating where both conditions are true simultaneously.
• **Simple Structure**: They are easier to handle algebraically and are often the starting point for solving more complex systems like quadratic or non-linear equations.

Understanding linear equations is essential for analyzing and predicting trends and for developing models in real-life applications.