Problem 88 - Algebra 1

Question

Use substitution to solve the system. $$\begin{aligned}&4 x-y=5\\\&2 x+4 y=15\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \( ( x, y) = (35/18, 5/9) \)

Step 1: Rearrange the first equation

The first step is to rearrange the first equation \(4x - y = 5\) to express y in terms of x. This can be done by adding y to both sides and subtracting 5 from both sides: \( y = 4x - 5 \)

Step 2: Substitute \(y\) in the second equation

Substitute \(y = 4x - 5\) in the second equation \(2x + 4y = 15\), to get an equation only in terms of x: \(2x + 4(4x - 5) = 15\).

Step 3: Solve for \(x\)

Simplify the equation to solve for \(x\): \(2x + 16x - 20 = 15\), then simplify further to get \(18x = 35\), which can be solved for \(x\) by dividing both sides by 18, to get \(x = 35 / 18\).

Step 4: Substitute \(x\) back to get value for \(y\)

Substitute \(x = 35 / 18\) back into equation \(y = 4x - 5\) to solve for \(y\), which gives \(y = 4(35/18) - 5 = 5 / 9\).

Step 5: Interpret the solution

The solution to the system of equations is the point \(( x, y) = (35/18, 5/9)\), which means that both of the given equations would be satisfied at this point.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. These equations work together and intersect at points that satisfy all equations in the system. Solving a system of equations involves finding the values of the variables that make all the equations true simultaneously.

• Types of systems: Systems can be consistent (having at least one solution) or inconsistent (having no solution).
• Solution methods: There are several ways to solve systems, including substitution, elimination, graphing, and matrices. The method chosen often depends on the complexity of the equations.

In this exercise, we are focusing on solving the system of equations using the substitution method. This method is particularly useful when one of the equations can easily be solved for one variable.

Solving Equations

Solving equations is the process of finding the values of variables that satisfy a given equation. When solving a single equation, you manipulate it to isolate the variable, leading to its solution. In a system of equations, the goal is to manipulate the system such that you find values that satisfy all equations simultaneously.

• Substitution method: This involves solving one equation for a variable and substituting this expression into another equation. This reduces the system to a single equation with one variable.
• Example from exercise: We solved the equation \(4x - y = 5\) for \(y\), getting \(y = 4x - 5\). This expression for \(y\) was then substituted into the second equation, \(2x + 4y = 15\), reducing it to a single equation in terms of \(x\).

This process simplifies the system into something more manageable and allows you to find the values of variables step-by-step.

Algebra 1

Algebra 1 is a foundational mathematics course that introduces students to the basic concepts of algebra. Understanding algebra is crucial as it provides the groundwork for advanced mathematics and various applications in science and engineering.

• Key Topics: Core topics in Algebra 1 include solving equations, graphing linear equations, understanding inequalities, and working with polynomials and quadratic equations.
• Importance of learning: Mastering these skills is essential as they are used in more advanced mathematics and in a variety of real-world scenarios.

Algebra teaches students to think critically and develop problem-solving skills. In the context of solving systems of equations, students apply their algebraic understanding to determine how different equations relate and interact. The substitution method covered in this exercise is one of the first steps students take in understanding systems, highlighting its role in Algebra 1.