Problem 78 - Precalculus

Question

If you graph the lines $$ \begin{array}{l} x-50 y=100 \\ x-48 y=-98 \end{array} $$ they appear to be parallel lines. However, there is a unique solution. Explain how this might be possible.

Step-by-Step Solution

Answer

The lines are coincident; they have the same slope and intersect at \((-4850, -99)\).

Step 1: Write the system of equations

We have the following system of linear equations: \[ x - 50y = 100 \] \[ x - 48y = -98 \] These are two linear equations in two variables \(x\) and \(y\).

Step 2: Align the equations for subtraction

To see if the lines really intersect or not, we can subtract one equation from the other. Align the equations for subtraction:\[ \begin{align*} (x - 50y) &= 100 \quad(1)\ (x - 48y) &= -98 \quad(2) \end{align*} \]

Step 3: Subtract the equations

Subtract Equation (2) from Equation (1):\[ (x - 50y) - (x - 48y) = 100 - (-98) \] Simplifying, we get:\[ -50y + 48y = 198 \]Which simplifies further to:\[ -2y = 198 \]

Step 4: Solve for \(y\)

Solve the equation \( -2y = 198 \) for \(y\): \[ y = \frac{198}{-2} \]\[ y = -99 \]

Step 5: Substitute \(y\) back to find \(x\)

Now that we have \(y = -99\), substitute it back into one of the original equations to find \(x\). Let's use the first equation:\[ x - 50(-99) = 100 \]This simplifies to:\[ x + 4950 = 100 \]Which gives:\[ x = 100 - 4950 \]\[ x = -4850 \]

Step 6: Verify the solution

Substitute \(x = -4850\) and \(y = -99\) into the second equation to verify:\[ x - 48y = -98 \]\[ -4850 - 48(-99) = -98 \]Simplify:\[ -4850 + 4752 = -98 \]\[ -98 = -98 \]The solution \(x = -4850\), \(y = -99\) satisfies both equations, confirming that it is correct.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed. These equations are usually in the format \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. The entire relationship between \( x \) and \( y \) in the equation defines a line. A single linear equation with two variables represents all the points on one specific line on a graph.

• The coefficients \( a \) and \( b \) determine the slope of the line.
• The constant \( c \) determines the y-intercept, where the line crosses the y-axis.

In our problem, both equations \( x - 50y = 100 \) and \( x - 48y = -98 \) are linear equations that, when solved together, describe the relationship and intersection of two lines.

Solution of a System

The solution of a system of equations refers to the set of values that satisfy all the equations in the system simultaneously. For two linear equations, this solution is the point where the lines represented by these equations intersect. Finding the solution involves determining the values of \( x \) and \( y \) that make both equations true.

• To solve the system, you can use methods like substitution, elimination, or graphing.
• Substitution involves solving one equation for a variable and then substituting that expression in the other equation.
• Elimination (or addition method) involves adding or subtracting equations to eliminate one of the variables.

In our exercise, we find the solution by aligning the equations and subtracting them, leading us to calculate the unique solution where both equations intersect: \( x = -4850 \) and \( y = -99 \).

Intersection of Lines

The intersection of lines is where two lines cross each other on a graph, representing a point that satisfies both lines' equations simultaneously. In geometric terms, this point of intersection is where the two linear paths meet.

• Geometrically, this point indicates the solution to the system of equations formed by the two lines.
• Algebraically, solving the system allows you to find the exact coordinates of the intersection.
• If lines do intersect, their solutions for \( x \) and \( y \) will be the same for both equations.

Our example reveals a solution, even though initially the lines seem parallel. The computations show that these lines do indeed intersect at the point \((-4850, -99)\).

Parallel Lines

Parallel lines have the same slope and will never intersect. They remain equidistant from each other regardless of how far they are extended. For two linear equations to represent parallel lines, their equations should differ only in their constant term when rearranged in slope-intercept form \( y = mx + c \).

• Two lines are parallel if they have identical slopes but different y-intercepts.
• Parallel lines are visually confirmed if graphed on a coordinate plane and never cross.

In the original exercise, the lines visually appear parallel. However, numerical calculations reveal they are not parallel because they intersect at a distinct point, proving that they don't maintain parallel consistency. Thus, our algebraic approach ensures an accurate understanding of the lines' relationship.