Problem 20 - Algebra and Trigonometry

Question

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 3. $$\left\\{\begin{aligned} 4 x+2 y &=16 \\ x-5 y &=70 \end{aligned}\right.$$

Step-by-Step Solution

Answer

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

Step 1: Understand the System of Equations

We are given a system of two linear equations. The first equation is \(4x + 2y = 16\) and the second equation is \(x - 5y = 70\). Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously, or determine if there is no solution or infinitely many solutions.

Step 2: Solve One Equation for One Variable

Start by solving one of the equations for one variable. Let's solve the second equation for \(x\): \(x = 5y + 70\).

Step 3: Substitute the Expression into the Other Equation

Substitute the expression we found for \(x\) into the first equation:\[4(5y + 70) + 2y = 16\].

Step 4: Simplify and Solve for y

Expand and simplify the substituted equation:\(20y + 280 + 2y = 16\)Combine like terms:\(22y + 280 = 16\).Subtract 280 from both sides:\(22y = -264\).Finally, divide by 22:\(y = -12\).

Step 5: Substitute Back to Find x

Use the value of \(y\) to find \(x\) using the expression from Step 2:\(x = 5(-12) + 70\), which simplifies to:\(x = -60 + 70\). So, \(x = 10\).

Step 6: Verify the Solution

Verify that \(x = 10\) and \(y = -12\) satisfies both original equations:First equation: \(4(10) + 2(-12) = 40 - 24 = 16\) (True).Second equation: \(10 - 5(-12) = 10 + 60 = 70\) (True).Both equations are satisfied, confirming that the solution is correct.

Key Concepts

Solving Systems of Equations

A system of equations consists of two or more equations sharing the same set of variables. In this type of mathematical problem, our objective is to find the particular values of the variables that satisfy all equations simultaneously. For instance, in the set of equations provided, the task is to determine the values of \(x\) and \(y\) such that both equations are true at the same time.
To solve systems of equations, there are several methods commonly used:

• Substitution: Solving one equation for one variable and substituting the result into another equation.
• Elimination: Combining equations to eliminate a variable, making it easier to solve for the remaining variable.
• Graphing: Drawing the equations on a graph to identify where they intersect, although this is less precise for exact solutions.

Understanding the nature of the solutions is crucial. If the system of equations is consistent and independent, there will typically be a single unique solution. If it is consistent and dependent, there will be infinitely many solutions.

Substitution Method

The substitution method is a popular technique for solving systems of linear equations. This approach involves solving one of the equations for one of the variables and then substituting this expression into the other equation.
Through this process, we effectively reduce a system of two equations into a single equation with one variable, simplifying the problem. In our example, the second equation \(x - 5y = 70\) was rearranged to express \(x\) in terms of \(y\), giving us \(x = 5y + 70\).
This new expression was substituted into the first equation \(4x + 2y = 16\), allowing us to focus solely on the variable \(y\). The substitution method is especially useful when one of the equations is easily manipulated to isolate a variable. It's a straightforward and commonly used method in linear algebra.

Linear Equations

Linear equations are equations that produce a straight line when graphed on a coordinate plane. They have the general form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. Each term in the equation is either a constant or the product of a constant and the first power of a variable.
In this system of equations:

• The first equation, \(4x + 2y = 16\), and
• The second equation, \(x - 5y = 70\)

represent two lines in the coordinate plane. Linear equations like these can have:

• A single intersection point, representing a unique solution.
• No intersection, if they are parallel, indicating no solutions.
• Overlapping lines, which indicate infinitely many solutions, as they coincide in every point.

Recognizing and understanding the structure of linear equations is vital for effectively solving them.

Solution Verification

Once a solution for a system of equations is found, it is critical to verify it by substituting the solution back into the original equations. Verification ensures that no mistakes were made during calculations and that the solution meets the original conditions posed by the equations.
For our specific problem, we determined that \(x = 10\) and \(y = -12\) satisfy both equations. When substituting \(x = 10\) and \(y = -12\) back into the first equation, \(4(10) + 2(-12)\) simplifies to \(16\), confirming the truth of the equation. Similarly, substituting into the second equation, \(10 - 5(-12)\) results in \(70\), further validating our solution.
By taking these steps, we can be confident that our solution is both accurate and reliable.