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Chapter 3: Problem 22

Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If \(a\) system has no solution, state this. $$ \begin{aligned} 4 x-y &=9 \\ x-3 y &=16 \end{aligned} $$

Short Answer

Expert verified
The solution is the point (1, -5).

Step by step solution

01

- Rewrite equations in slope-intercept form

To graph the equations, convert them to the slope-intercept form, which is \(y = mx + b\). For the first equation, rewrite it as follows: \(4x - y = 9\)\[-y = -4x + 9\]y = 4x - 9For the second equation, rewrite it as follows: \(x - 3y = 16\)x - 16 = 3y\[y = \frac{1}{3}x - \frac{16}{3} \].
02

- Graph the equations

Graph the equations on the same coordinate system. The first equation is \(y = 4x - 9\), which has a slope of 4 and a y-intercept of -9. The second equation is \(y = \frac{1}{3}x - \frac{16}{3}\), which has a slope of \( \frac{1}{3}\) and a y-intercept of \(-\frac{16}{3}\).
03

- Find the intersection point

Determine where the lines intersect by solving the system of equations. Set the equations equal to each other:\[4x - 9 = \frac{1}{3}x - \frac{16}{3}\]Multiply both sides by 3 to clear the fraction:\[12x - 27 = x - 16\]Collect like terms:\[11x = 11\]Divide by 11:\[x = 1\]Substitute \(x = 1\) back into one of the original equations to find \(y\):\[4(1) - y = 9\]\[4 - y = 9\]\[y = -5\].
04

- Verify the solution

Verify the solution by substituting \(x = 1\) and \(y = -5\) into both original equations.For \(4x - y = 9\): \[4(1) - (-5) = 9\]\[4 + 5 = 9\], which is true.For \(x - 3y = 16\)\[1 - 3(-5) = 16\]\[1 + 15 = 16\], which is true.
05

- State the solution

Since both equations are satisfied by \(x = 1\) and \(y = -5\), the solution to the system is the point (1, -5).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
To solve a system of equations graphically, it is helpful to start by converting each equation into slope-intercept form. This form is represented as: \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. Starting with the slope-intercept form helps you easily draw each line on a coordinate plane. For the given equations, we convert them as follows:
  • \bm{Equation 1:} \(4x - y = 9\)\- y = -4x + 9\y = 4x - 9\
  • \bm{Equation 2:} \(x - 3y = 16\)\x = 3y + 16\y = \frac{1}{3}x - \frac{16}{3}\
During this step, you rearrange terms to isolate \(y\) and identify your slope and y-intercept for each equation.
graphing linear equations
Once your equations are in slope-intercept form, you can graph them on a coordinate plane. This step involves:
  • \bm{Identifying the y-intercept (b):} The point where the line crosses the y-axis.
  • \bm{Using the slope (m):} To determine the rise over run from the y-intercept. For the line \(y = 4x - 9\), the slope is 4, which means for each step of 1 unit right, the line moves 4 units up.
  • \bm{For the second line \(y = \frac{1}{3}x - \frac{16}{3}\):} The slope is \( \frac{1}{3}\), meaning the line moves up 1 unit for every 3 units to the right.
Mark the points and use a ruler to draw straight lines extending in both directions. Ensure both lines are accurately plotted as their intersection will determine the solution.
intersection point
The intersection point of the graph's lines represents the solution to the system of equations. To find this point graphically:
  • \bm{Set equations equal:} \(4x - 9 = \frac{1}{3}x - \frac{16}{3}\)
  • \bm{Clear fractions:} Multiply both sides by 3 \( \rightarrow 12x - 27 = x - 16 \)
  • \bm{Solve for x:} Combine like terms \( \rightarrow 11x = 11 \)
  • \bm{Find x:} \( x = 1 \)
  • \bm{Substitute x back:} Into either original equation to find y \(4(1) - 9 = -5 \)
  • \bm{Solution:} The intersection point and solution is (1, -5)
This point shows where both equations yield the same x and y values.
solution verification
Always verify your solution by substituting back into the original equations. This ensures the solution satisfies both equations.
For our solution \(x = 1\) and \(y = -5\):
  • \bm{Equation 1:} \(4(1) - (-5) = 9 \)\4 + 5 = 9\
  • \bm{Equation 2:} \(1 - 3(-5) = 16 \)\1 + 15 = 16\
Both equations hold true, confirming the solution. This step is crucial to detect any graphing errors or miscalculations, ensuring confidence in your solution鈥檚 accuracy.

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