Problem 19 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} 4 x+y=11 \\ 2 x+5 y=1 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((x, y) = (3, -1)\).

Step 1: Solve the First Equation for y

Start with the first equation: \( 4x + y = 11 \). Rearrange it to express \( y \) in terms of \( x \): \( y = 11 - 4x \). This is the expression we will substitute into the second equation.

Step 2: Substitute y in Second Equation

Substitute the expression for \( y \) from Step 1 into the second equation: \( 2x + 5y = 1 \). This gives us: \( 2x + 5(11 - 4x) = 1 \).

Step 3: Simplify the Substituted Equation

Expand the equation: \( 2x + 55 - 20x = 1 \). Combine like terms to get \( -18x + 55 = 1 \).

Step 4: Solve for x

Rearrange the equation to isolate \( x \): \( -18x = 1 - 55 \), which simplifies to \( -18x = -54 \). Divide both sides by \( -18 \) to find \( x = 3 \).

Step 5: Substitute x back to Solve for y

Substitute \( x = 3 \) back into the expression \( y = 11 - 4x \) from Step 1. Calculate \( y = 11 - 4(3) = 11 - 12 = -1 \).

Step 6: Write the Solution as a Point

The solution to the system of equations is the point \((x, y) = (3, -1)\).

Key Concepts

System of Equations

A system of equations consists of multiple equations that share the same variables. The goal is to find a set of values for these variables that satisfy all the equations simultaneously. To understand this, imagine two different lines on a graph representing each equation. The solution to the system is where the two lines intersect. If they do not intersect, the system has no solution. If they overlap completely, there are infinitely many solutions. There are different methods to solve systems of equations, such as graphing, substitution, and elimination. Each method has its own advantages depending on the situation. For instance, substitution is quite useful when one equation can easily be rearranged to express one variable in terms of the others, as shown in our exercise. Understanding this concept helps in solving a wide variety of problems where relationships between variables are described using equations.

Solving Linear Equations

Linear equations are foundational in algebra. They represent straight lines and have no variables with exponents greater than one. Understanding how to solve linear equations equips you with skills to tackle more complex mathematical problems. When solving linear equations, your primary goal is to isolate the variable you are trying to solve for. This typically involves performing operations that "undo" what has been done to the variable, such as:

• Adding or subtracting terms on both sides to move terms involving the variable to one side of the equation and constant terms to the other side.
• Multiplying or dividing both sides of the equation by a number to solve for the variable.

These techniques were demonstrated in the step-by-step solution above, where terms were rearranged and simplified to solve for each variable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like x or y), and operations (such as plus or minus). They are crucial parts of algebra that allow us to generalize mathematical laws and relationships. In our exercise, algebraic expressions were used to express one variable in terms of another. This kind of manipulation is often the first step in solving a system of equations by substitution. The key process involves simplifying expressions by combining like terms and applying arithmetic operations:

• Combine terms that have the same variable to simplify the expression.
• Distribute numbers across parentheses to expand expressions, as shown when solving the system of equations.

Mastering algebraic expressions will help you not only in solving equations but also in performing more advanced calculations in future math studies.