Problem 34 - Algebra and Trigonometry

Question

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{aligned} 4 x+2 y &=16 \\ x-5 y &=70 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \((x, y) = (10, -12)\).

Step 1: Identify the Equations in the System

The given system of equations is: 1. \(4x + 2y = 16\)2. \(x - 5y = 70\).

Step 2: Solve One Equation for a Variable

We can solve the second equation for \(x\). To isolate \(x\), add \(5y\) to both sides of the equation: \[ x = 70 + 5y \]

Step 3: Substitute into the Other Equation

Substitute the expression for \(x\) from Step 2 into the first equation:\(4(70 + 5y) + 2y = 16\).Distribute the \(4\): \(280 + 20y + 2y = 16\).Combine like terms: \(280 + 22y = 16\).

Step 4: Solve for the Isolated Variable

Subtract \(280\) from both sides of the equation to isolate the \(y\) term:\[ 22y = 16 - 280 \]\[ 22y = -264 \]Divide both sides by 22:\[ y = \frac{-264}{22} = -12 \]

Step 5: Solve for the Other Variable

Use the expression for \(x\) from Step 2 with the value of \(y\) found in Step 4:\[ x = 70 + 5(-12) \]\[ x = 70 - 60 \]\[ x = 10 \]

Step 6: Verify the Solution

Substitute \(x = 10\) and \(y = -12\) back into both original equations:1. \(4(10) + 2(-12) = 40 - 24 = 16\) (True).2. \(10 - 5(-12) = 10 + 60 = 70\) (True).Since both equations are satisfied, the solution is verified.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share common variables. In our example, we are dealing with the following system of equations:

• The first equation is: \(4x + 2y = 16\).
• The second equation is: \(x - 5y = 70\).

Every solution to this system must satisfy both equations simultaneously. Finding a solution means determining the values of the variables \(x\) and \(y\) that make both equations true at the same time. We can typically encounter different types of solutions for systems of equations.

• A unique solution, which is one specific pair of \(x\) and \(y\) values.
• No solution, meaning the equations represent parallel lines that never intersect.
• Infinitely many solutions, where the equations represent the same line, and thus intersect everywhere.

In our problem, the system possesses a unique solution.

Substitution Method

The substitution method is a common technique to solve systems of linear equations. It involves expressing one variable in terms of another from one equation, and then substituting that expression into another equation. This method is particularly useful when one equation is easily solvable for one variable.
Steps for applying the substitution method include:

• Solve one of the equations for one variable. In this case, we solve the second equation for \(x\): \[ x = 70 + 5y \]
• Substitute the expression obtained into the other equation, turning it into a single-variable equation. This step allows us to find the value of the other variable. For our equations, substitute \(x = 70 + 5y\) into the first equation: \[ 4(70 + 5y) + 2y = 16 \]
• Simplify and solve the resulting equation for \(y\).
• Once \(y\) is found, return to the expression used for substitution to solve for the other variable \(x\).

This method ensures both equations in the system are satisfied.

Solving for Variables

When solving for variables in a system of equations using substitution, the aim is to isolate each variable step-by-step. After substituting the expression for \(x\) into the other equation, we simplify to get:

• Distribute: \[ 280 + 20y + 2y = 16 \]
• Combine like terms: \[ 280 + 22y = 16 \]
• Isolate \(y\) by subtracting 280: \[ 22y = 16 - 280 \]
• Simplify: \[ 22y = -264 \]
• Divide by 22: \( y = -12 \).

With \(y\) determined, find \(x\) using the earlier expression \(x = 70 + 5y\). Substitute \(y = -12\) in:

• \[ x = 70 + 5(-12) \]
• Simplify: \[ x = 70 - 60 \]
• \[ x = 10 \]

Verification can be helpful, so substitute back into the original equations to ensure both are satisfied with \(x = 10\) and \(y = -12\). This confirms correctness of the solution.