Problem 21 - Algebra 1: Concepts and Skills

Question

Estimate the solution of the linear system graphically. Then check the solution algebraically. $$ \begin{array}{r} {x-y=1} \\ {5 x-4 y=0} \end{array} $$

Step-by-Step Solution

Answer

The solution of the system of equations is (1, 0).

Step 1: Convert to Slope-Intercept Form

Rewrite the given equations in the form y = mx + b. For the first equation, add y to both sides to get \(x = y + 1\) or \(y = x - 1\). For the second equation, add 4y to both sides and divide by 4 to get \(y = 5/4x\). Now, we have two equations: \(y = x - 1\) and \(y = 5/4x\).

Step 2: Graph The Equations

Graph the equations \(y = x - 1\) and \(y = 5/4x\) on the same set of axes. The solution to the system of equations is the point where these two lines intersect.

Step 3: Find The Intersection Point

The intersection point is found graphically to be the point (1, 0). This solution can now be verified algebraically.

Step 4: Verify The Solution Algebraically

Substitute x = 1 and y = 0 into the original equations. For the first equation, we get 1 - 0 = 1, which is true. For the second equation, we get 5*1 - 4*0 = 5, which is also true. Therefore, our graphical solution of (1, 0) is confirmed algebraically.

Key Concepts

Graphical Solution

A graphical solution involves plotting equations on a coordinate plane to find their intersecting point visually. In our given problem, we need to graph the linear equations to estimate where they meet.

• Firstly, convert the equations to slope-intercept form, which makes them easier to graph.
• Then, draw each line on a grid using the intercepts and slope calculations.
• Observe where they cross each other. This intersection represents the solution to the system.

This method visually reveals the solution by showing us the point where both lines coincide on the graph.

Slope-Intercept Form

The slope-intercept form is a way to express a linear equation as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form makes it straightforward to graph lines. For the equation \(x - y = 1\):

• Rearrange to \(y = x - 1\).
• Here, the slope \(m = 1\) and the y-intercept \(b = -1\).

For the equation \(5x - 4y = 0\):

• Rearrange to \(y = \frac{5}{4}x\).
• Here, the slope \(m = \frac{5}{4}\) and the y-intercept \(b = 0\).

These transformations ease the graphing process, highlighting the direction and starting point of each line.

Intersection Point

The intersection point of two lines is crucial because it represents the solution to their system of equations. In our exercise, graph each transformed line, \(y = x - 1\) and \(y = \frac{5}{4}x\), on the same set of axes.

• Observe where the lines cross to find the intersection point.
• In this exercise, the lines intersect at \((1, 0)\).
• This coordinate is significant because it satisfies both linear equations simultaneously.

By identifying this point, we determine the values of \(x\) and \(y\) that make both original equations true.

Algebraic Verification

Once we have estimated the solution graphically, algebraic verification confirms its accuracy. We know our intersection point is \((1, 0)\). We'll substitute \(x = 1\) and \(y = 0\) back into the original equations to check:For the first equation \(x - y = 1\):

• Plugging the values in, it simplifies to \(1 - 0 = 1\).
• This statement is true, verifying the first equation.

For the second equation \(5x - 4y = 0\):

• Substitute in to get \(5 \times 1 - 4 \times 0 = 5\).
• This is also true, confirming the second equation.

Each equation must be true for the solution to be verified. Here, both equations hold, affirming that \((1, 0)\) is indeed the solution to the system. This dual check of graphical observation and algebraic confirmation strengthens our solution's reliability.