Problem 45 - Precalculus Mathematics for Calculus

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 6. $$\left\\{\begin{array}{l}0.4 x+1.2 y=14 \\\12 x-5 y=10\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has a single solution: \((5, 10)\).

Step 1: Simplify the first equation

The first equation is given as \(0.4x + 1.2y = 14\). We will multiply the entire equation by 10 to eliminate the decimals: \(4x + 12y = 140\).

Step 2: Rearrange the equations

Now we have two equations: \(4x + 12y = 140\) and \(12x - 5y = 10\). We will work with these equations to solve the system.

Step 3: Eliminate variable 'x'

To eliminate the variable 'x', we can multiply the first equation by 3 and the second equation by 1, making the coefficients of 'x' in both equations equal 12.The equations become:\[\begin{align*}3(4x + 12y) &= 3(140) \12(4x) - 5y &= 10\end{align*}\]Which simplifies to:\[\begin{align*}12x + 36y &= 420 \12x - 5y &= 10\end{align*}\]

Step 4: Subtract the equations

Subtract the second equation from the first equation to eliminate 'x':\[(12x + 36y) - (12x - 5y) = 420 - 10\]This results in:\[41y = 410\]

Step 5: Solve for 'y'

Divide both sides of the equation by 41 to solve for 'y':\[y = \frac{410}{41} = 10\]

Step 6: Substitute back to find 'x'

Substitute \(y = 10\) back into the first equation \(4x + 12y = 140\):\[4x + 12(10) = 140\]Simplify and solve for 'x':\[4x + 120 = 140 \4x = 20 \x = 5\]

Step 7: Write the solution as an ordered pair

With \(x = 5\) and \(y = 10\), the solution in ordered pair form is \((5, 10)\).

Key Concepts

Solving Systems of Linear Equations

When you encounter a system of linear equations, your goal is to find the values of the variables that make all equations true at the same time. These systems can have different types of solutions, depending on the equations:

• One Solution: This happens when lines intersect at precisely one point. The system is known as consistent and independent.
• Infinitely Many Solutions: This occurs when the lines coincide, meaning they lie on top of each other and share all points. The system is consistent and dependent.
• No Solution: If the lines are parallel and never meet, the system has no solution and is inconsistent.

Recognizing the solution type helps you decide the best method to solve the system, like substitution, elimination, or graphing. Be prepared to express solutions in ordered pair form like (\(x, y\)), representing the intersection point.
This gives you a clear picture of where equations "agree" in their solutions.

Elimination Method

The elimination method is a popular technique to solve systems of equations algebraically. This method aims to eliminate one variable, making it easier to solve for the other. Here's how it works:

• Align Equations: Start by writing both equations neatly stacked, ensuring all variables are lined up.
• Adjust Coefficients: Multiply one or both equations to make the coefficient of one variable the same in both equations.
• Eliminate a Variable: Add or subtract the equations to remove one variable, leaving an equation with one variable.
• Solve for Remaining Variable: With one variable gone, solve the remaining equation for the other variable.

In our example, we used multiplication to make the 'x' coefficients equal in both equations, enabling us to subtract one equation from the other and eliminate 'x'. After simplifying, we found the value of 'y', which we could then substitute back to find 'x'.
Practicing this method can greatly enhance your algebra problem-solving skills.

Ordered Pair Solutions

An ordered pair solution represents the x and y values that satisfy a system of linear equations. It’s written as (\(x, y\)) where \(x\) and \(y\) are the values of the variables determined by solving the equations.

• In algebra, ordered pairs are not just a convenient notation; they provide a geometric interpretation as well. When solving equations, the ordered pair tells you the intersection of the lines formed by the equations.
• The pair (5, 10) from our example means that when \(x = 5\) and \(y = 10\), both equations are satisfied.

This solution confirms it is exactly where our two lines cross each other on a graph. Accurately writing and interpreting ordered pairs is crucial in both algebraic solutions and real-life situations where such equations are used.