Problem 32 - Thinking Mathematically

Question

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}3 x-7 y=13 \\ 6 x+5 y=7\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x=2\) and \(y=-1\).

Step 1: Multiply Equation

To apply the addition method, the equations should be arranged in a way that adding them will eliminate one variable. The best approach is to multiply the second equation by 7 and the first equation by 5 so that we get two new equations to work with: \(15x-35y=65\) and \(42x+35y=49\)

Step 2: Add the Transformed Equations

After transformation, the equations can be added together. Adding the resulting equations: \(15x-35y+42x+35y=65+49\). This simplifies to \(57x=114\).

Step 3: Solve for x

By simplifying the above equation, you can find the value of x. Divide both sides of the equation by 57 to isolate x. This results in \(x=2\).

Step 4: Substitute x into Original Equation

Substitute x=2 into the first original equation: \(3(2)-7y=13\), which simplifies to \(6-7y=13\).

Step 5: Solve for y

Re-arrange the equation in Step 4 to solve for y. Subtract 6 from both sides: \(-7y=13-6\), which simplifies to \(-7y=7\). Finally, divide both sides by -7 to find \(y=-1\).

Key Concepts

Linear Equations

Linear equations are fundamental components of algebra. They are equations of the first order, which means they have no exponents higher than one. This makes them straight-line graphs when plotted on a coordinate grid. In a linear equation like\( ax + by = c \), every term is either a constant or a product of a constant and a single variable.Linear equations often involve:

• Variables such as \( x \) and \( y \)
• Constants like \( a, b, \) and \( c \)
• Operations of addition, subtraction, or multiplication by a constant

Linear equations are simple yet powerful building blocks for understanding more complex algebraic ideas. They make up the basis of systems of equations, where multiple linear equations interact and intersect.

Systems of Equations

A system of equations is essentially a set of two or more equations with the same variables. Solving these systems means finding a set of values for the variables that satisfies all equations simultaneously. In our original exercise, the system of equations involved two linear equations:

• \(3 x - 7 y = 13 \)
• \(6 x + 5 y = 7 \)

Such systems can represent various real-world situations where different conditions need to be satisfied together, such as optimizing cost and materials in manufacturing.To solve these systems, various methods exist:

• Graphing
• Substitution
• Addition (or elimination)

The addition method combines the equations to eliminate one variable, making one equation with only one unknown. This process simplifies solving and is effective for systems with multiple linear equations.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating them. It's a unifying thread of almost all of mathematics and involves solving for unknown values in an equation by performing different operations. It uses various symbols—often letters—and involves the following key components:

• Expressions, which are combinations of symbols that represent numbers
• Equations, which express equality between two expressions
• Variables that hold unknown values to solve for

Algebra provides the foundation for most areas of mathematics. It allows us to solve linear equations, systems of equations, and much more. By understanding the properties of equality and variable manipulation, equations like \(15x - 35y = 65\) can be worked through systematically to find solutions.

Solving Equations

When we talk about solving equations, we usually mean finding the value of variables that make the equation true. The process involves several steps:

Setting Up the Equations

This involves expressing the problem as equations, like setting up \(3 x - 7 y = 13\).

Manipulating the Equations

This can involve:

• Rearranging terms
• Multiplying for simplifying
• Adding or subtracting equations to eliminate variables

Finding Values

Once simplified, solve for one variable at a time. For instance, finding \(x = 2\).

Verification:

Always insert your solutions back into the original equations to verify your results.

Practice

Regular practice is essential. Solving different systems hones your ability to see patterns and apply appropriate methods rapidly. Solving equations, especially linear ones, is crucial for critical thinking and problem-solving in mathematics and real-world applications.