Problem 5 - Algebra and Trigonometry

Question

\(5-8\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{aligned} x-y &=1 \\ 4 x+3 y &=18 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution to the system is \(x = 3\) and \(y = 2\).

Step 1: Express one variable in terms of the other

From the first equation, \(x - y = 1\), we can express \(x\) in terms of \(y\). Rewrite the equation as \(x = y + 1\). This allows us to substitute \(x\) in the second equation.

Step 2: Substitute expression into the second equation

Substitute \(x = y + 1\) into the second equation \(4x + 3y = 18\). This gives us: \(4(y + 1) + 3y = 18\). Simplify this equation to solve for \(y\).

Step 3: Simplify and solve for y

Distribute the 4 in the equation: \(4y + 4 + 3y = 18\). Combine like terms to get \(7y + 4 = 18\). Then, subtract 4 from both sides to obtain \(7y = 14\). Finally, divide by 7 to find \(y = 2\).

Step 4: Substitute back to find x

Substitute \(y = 2\) back into the expression for \(x\), which is \(x = y + 1\). This gives \(x = 2 + 1\). Simplify to find \(x = 3\).

Step 5: Write the solution to the system

The system is solved, and the solution is \((x, y) = (3, 2)\). This means that \(x = 3\) and \(y = 2\) satisfy both equations in the system.

Key Concepts

System of Equations

A system of equations consists of two or more equations involving the same set of variables. When solving a system of equations, we seek values for the variables that will satisfy all of the equations simultaneously. In our exercise, the system is made up of two equations:

• \(x - y = 1\)
• \(4x + 3y = 18\)

These equations define constraints on variables \(x\) and \(y\). The goal is to find a pair \((x, y)\) that makes both equations true at the same time.There are several methods to solve systems of equations, such as graphing, substitution, and elimination. Here, we focus on the substitution method. It’s especially useful when one of the equations is easily solvable for one variable, as it allows us to substitute that expression into the other equation, reducing the number of variables.

Linear Equations

Linear equations are equations of the first degree, which means the highest exponent of the variable is one. These equations, when graphed, produce a straight line. In a two-variable context like our problem, each equation in a system represents a line in a two-dimensional space.In the given system, both \(x - y = 1\) and \(4x + 3y = 18\) are linear equations. The significance of linear equations is that they can be put into the standard form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. This form is convenient for identifying solutions graphically through intersections. However, our problem uses algebraic manipulation to find where these lines intersect, representing the solution to the system of equations.

Solving Equations Step-by-Step

The substitution method for solving systems of equations relies on expressing one variable in terms of another, allowing us to solve with a clear plan.First, take one of the equations and isolate a variable. For instance, from \(x - y = 1\), we solve for \(x\):\[ x = y + 1 \]Next, substitute this expression into the other equation. Plug \(x = y + 1\) into \(4x + 3y = 18\):\[ 4(y + 1) + 3y = 18 \]Simplifying and solving for \(y\), distribute and combine like terms:\[ 4y + 4 + 3y = 18 \]This reduces to:\[ 7y + 4 = 18 \]Subtract 4 and divide by 7 to find:\[ y = 2 \]Finally, substitute back to solve for \(x\):\[ x = 2 + 1 = 3 \]The solution \((x, y) = (3, 2)\) is the intersection point of the two lines, satisfying both original equations.