Problem 18 - Algebra 2

Question

Solve each system of equations by using elimination. \(u+v=7\) \(2 u+v=11\)

Step-by-Step Solution

Answer

The solution is \( u = 4 \) and \( v = 3 \).

Step 1: Align the Equations

Write down the given system of equations, aligning corresponding variables and constants:1. \( u + v = 7 \) 2. \( 2u + v = 11 \)

Step 2: Subtract to Eliminate v

Subtract the first equation from the second to eliminate \( v \):\((2u + v) - (u + v) = 11 - 7\)This simplifies to:\(u = 4\)

Step 3: Substitute to Find v

Substitute \( u = 4 \) back into the first equation \( u + v = 7 \):\(4 + v = 7\)Solve for \( v \):\(v = 3\)

Step 4: Write the Solution

The solution to the system of equations is \( u = 4 \) and \( v = 3 \). This is written as the ordered pair \( (4, 3) \).

Key Concepts

Systems of Equations

When solving mathematical problems, you may come across a system of equations. A system of equations is a set of two or more equations that you deal with at the same time. These equations share variables. For example, in our original exercise, we have two equations: \( u + v = 7 \) and \( 2u + v = 11 \).

The main goal is to find the values of the variables, like \( u \) and \( v \), that satisfy all the equations at the same time. Solving systems of equations is like finding a sweet spot: It's where all the conditions from different equations are satisfied.

This is important in many real-world problems where, for example, you need to find the best solution between various constraints. Thus, systems of equations provide a way to model and solve these complex scenarios.

Substitution Method

Another common technique to solve systems of equations is the substitution method. Unlike elimination, which we used in the original exercise, substitution works by solving one equation for a variable and then replacing this variable in the other equation.

Here's how it works step-by-step:

• Solve one of the equations for one variable in terms of the other. For instance, if we solved \( u + v = 7 \) for \( v \), we'd get \( v = 7 - u \).
• Substitute this expression into the other equation. Replace \( v \) in \( 2u + v = 11 \) with \( 7 - u \), giving \( 2u + (7 - u) = 11 \).
• Solve the resulting single-variable equation to find the value of the variable. Here, solving yields \( u = 4 \).
• Substitute back to find the other variable. By replacing \( u \) in \( v = 7 - u \), you find \( v = 3 \).

Using substitution can often be more intuitive, especially when one equation is already solved for a particular variable. It is a complementary approach to elimination and useful in various situations.

Algebraic Solution

The algebraic solution involves using mathematical operations to find the unknowns in equations. In solving systems of equations, algebra helps break down the problem into manageable pieces.

For example, in the original exercise, subtraction was the key operation used. By subtracting one equation from another, we eliminated one of the variables \( v \), simplifying our equations. This is a crucial step as it leads us straight to the solution for \( u \).

Algebra serves as the underlying framework that allows us to manipulate equations using rules and properties, like the distributive, associative, and commutative properties. From these rules, we derive steps like adding, subtracting, multiplying, or dividing entire equations or specific terms within them.

An algebraic solution is vital because it translates challenging, complex problems into a more straightforward form. It is a fundamental concept required to progress in solving equations—you’re essentially solving a puzzle using math!

Linear Equations

Linear equations are the building blocks of systems of equations. They are called "linear" because they graph as straight lines.

Each linear equation has the general form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. In the original exercise, both equations like \( u + v = 7 \) and \( 2u + v = 11 \) are linear because their graphs would be straight lines when plotted.

The magic happens when you consider these lines together. If two lines intersect at a point, that point is the solution to the system of equations. This is true for any system of two linear equations with two unknowns, assuming a unique solution exists.

Understanding linear equations is essential because they form the basis for more complex topics in algebra, such as quadratic equations, inequalities, and calculus. Knowing how to solve linear equations is the first step in dealing with more advanced mathematical problems.