Problem 19 - Algebra 1: Concepts and Skills

Question

Estimate the solution of the linear system graphically. Then check the solution algebraically. $$ \begin{array}{r} {2 x-3 y=9} \\ {x=-3} \end{array} $$

Step-by-Step Solution

Answer

The solution to the system of equations is = \[x, y\]

Step 1: Graphical Solution Preparation

To solve the system graphically, first rewrite the equations in slope-intercept form \(y = mx + b\). The second equation is already in this form where \(x = -3\), which represents a vertical line. The first equation can be reformatted to: \(y = \frac{2}{3}x - 3\)

Step 2: Graphical Solution

Plot these two lines on a graph. The first line has a slope of \(\frac{2}{3}\) and y-intercept at -3. The second line is a vertical line crossing the x-axis at -3. Identify the point where these two lines intersect. This point represents the solution to the system of linear equations.

Step 3: Algebraic Solution

The algebraic solution involves substituting the \(x\) value from the second equation into the first equation. Doing so, we get: \(2(-3) - 3y = 9\). Solve the above equation to get the value of \(y\).

Step 4: Verification

After finding the values of \(x\) and \(y\), substitute them back into each of the original equations. Both equations must hold true with these values for the solution to be correct.

Key Concepts

Understanding a Linear System

A linear system consists of two or more linear equations. Each equation represents a straight line when graphed in the coordinate plane. The solution to the linear system is the point where all the lines intersect. In the given exercise, the goal is to find the solution for the system of the two linear equations:

• Equation 1: \(2x - 3y = 9\)
• Equation 2: \(x = -3\)

These equations can depict scenarios where two variables, typically \(x\) and \(y\), need to satisfy both equations simultaneously. By solving the system, we are finding a set of values for \(x\) and \(y\) that make both equations true.
In many cases, you can find this solution both graphically and algebraically.

Converting to Slope-Intercept Form

The slope-intercept form, expressed as \(y = mx + b\), is a way to represent linear equations that emphasizes the slope and y-intercept of the line. Converting equations to this form allows for easier graphing and understanding of the line's characteristics.
For the equation \(2x - 3y = 9\), converting to slope-intercept form involves solving for \(y\):

• Subtract \(2x\) from both sides: \(-3y = -2x + 9\)
• Divide every term by \(-3\): \(y = \frac{2}{3}x - 3\)

The resulting equation \(y = \frac{2}{3}x - 3\) reveals:

• The slope \(m\) is \(\frac{2}{3}\)
• The y-intercept \(b\) is \(-3\)

This format makes it straightforward to plot the line on a graph. The second equation \(x = -3\) is an exception—it's a vertical line and does not need conversion to slope-intercept form.

Finding the Solution Algebraically

To find the solution algebraically, we substitute the given value of \(x\) from one equation into the other. For the given system, the second equation provides \(x = -3\). Substitute this value into the first equation:
\(2(-3) - 3y = 9\)
This simplifies to:

• \(-6 - 3y = 9\)
• Add 6 to both sides: \(-3y = 15\)
• Divide by \(-3\): \(y = -5\)

Thus, the solution using algebraic methods is \(x = -3\) and \(y = -5\). To verify, plug \(x = -3\) and \(y = -5\) back into the original equations:

• In \(2x - 3y = 9\): \(2(-3) - 3(-5) = 9\), which simplifies to \(-6 + 15 = 9\).
• In \(x = -3\), the \(x\) value holds true as \(-3\).

Both checks confirm the solution is correct.