Problem 36 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}4 x-y=7 \\ 2 x-3 y=-9\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is the intersection point (3, 5).

Step 1: Write the Equations in Slope-Intercept Form

To graph these equations easily, we need to convert them into the slope-intercept form, which is \(y = mx + b\). First, solve the equation \(4x - y = 7\) for \(y\):\[y = 4x - 7\]Now, solve the equation \(2x - 3y = -9\) for \(y\):\[3y = 2x + 9 \y = \frac{2}{3}x + 3\]We now have the two equations: \(y = 4x - 7\) and \(y = \frac{2}{3}x + 3\).

Step 2: Graph the First Equation

Start by graphing the equation \(y = 4x - 7\). - The y-intercept \(b\) is -7, so plot the point (0, -7) on the y-axis.- The slope \(m\) is 4, which means rise 4 units for every 1 unit run (\(\frac{4}{1}\)). From (0, -7), move up 4 units and right 1 unit to plot the second point (1, -3).- Draw the line through (0, -7) and (1, -3).

Step 3: Graph the Second Equation

Now graph the equation \(y = \frac{2}{3}x + 3\).- The y-intercept \(b\) is 3, so plot the point (0, 3) on the y-axis.- The slope \(m\) is \(\frac{2}{3}\), meaning rise 2 units for every 3 units run. From (0, 3), move up 2 units and right 3 units to plot the second point (3, 5).- Draw the line through (0, 3) and (3, 5).

Step 4: Identify the Intersection Point

Observe where the two lines intersect on the graph. The intersection point is the solution to the system of equations. Look closely at the graph to find the point where both lines cross. This intersection point is the solution.

Step 5: Confirm the Solution by Substitution

After determining the intersection point from the graph, substitute the coordinates into the original equations to verify the solution.Graphically, the two lines meet at the point (3, 5). Substitute \(x = 3\) and \(y = 5\) into the original equations to confirm:1. For \(4x - y = 7\): \(4(3) - 5 = 12 - 5 = 7\).2. For \(2x - 3y = -9\): \(2(3) - 3(5) = 6 - 15 = -9\).Both conditions are satisfied, confirming the solution.

Key Concepts

Graphing Method

Graphing is a visual method to solve a system of linear equations. This method helps us find the exact point where two lines intersect on the graph, which represents the solution to the system. Here's how the graphing method works for solving systems of linear equations:

• Begin by transforming each equation into a graphable form, such as the slope-intercept form (\(y = mx + b\)).
• Plot each equation on the same coordinate graph using the determined slopes and y-intercepts.
• Once both lines are sketched, observe where they cross. This intersection point is the solution that satisfies both equations.

The strength of the graphing method lies in its simplicity and the visual understanding it provides, making it highly suitable for smaller systems and those who are visually inclined.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most straightforward and commonly used forms. Written as \(y = mx + b\), this form directly reveals two critical pieces of information about the line:- **Slope \(m\):** This tells us how steep the line is. It describes the rate at which \(y\) changes with respect to \(x\). Typically, the slope is expressed as \(\frac{\text{rise}}{\text{run}}\), where it signifies the vertical movement (rise) over horizontal movement (run). For example, in the equation \(y = 4x - 7\), the slope is 4, indicating a steep incline.- **Y-intercept \(b\):** This value shows where the line crosses the y-axis. It is the value of \(y\) when \(x = 0\). For instance, in the equation \(y = \frac{2}{3}x + 3\), the y-intercept is 3.Being able to quickly sketch these lines using the y-intercept and slope simplifies the visualization and solution of linear systems through graphing.

Intersection Point Verification

Once you have drawn the graphs of your equations and identified the intersection point visually, it is essential to verify this result mathematically to ensure accuracy. A graphical solution gives an approximate answer, while substitution provides exact confirmation.**How to Verify:**

• Determine the coordinates where the lines intersect. Let's say the found intersection is (3, 5).
• Substitute these \((x, y)\) values back into each of the original equations.
• Ensure both equations are true. For example, substituting into the first equation yields \(4(3) - 5 = 7\), which is correct. Verify the second equation similarly.

Substituting the intersection point into the original equations confirms the solution is correct. This step confirms both the visual and mathematical solutions align, assuring correctness.