Questions: Solve the system of equations by graphing. 4x + 3y = 59 -3x + 4y = 37 Use the graphing tool to graph the system.

Solve the system of equations by graphing. 4x + 3y = 59 -3x + 4y = 37 Use the graphing tool to graph the system.
Transcript text: Solve the system of equations by graphing. \[ \begin{array}{r} 4 x+3 y=59 \\ -3 x+4 y=37 \end{array} \] Use the graphing tool to graph the system.
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Solution

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Solution Steps

Step 1: Rewrite equations in slope-intercept form

To rewrite the equations in slope-intercept form (y = mx + b), we isolate y:

  • Equation 1: 4x + 3y = 59 3y = -4x + 59 y = (-4/3)x + 59/3

  • Equation 2: -3x + 4y = 37 4y = 3x + 37 y = (3/4)x + 37/4

Step 2: Identify key features for graphing

Now, we identify the slope (m) and y-intercept (b) for each equation:

  • Equation 1: y = (-4/3)x + 59/3 Slope (m) = -4/3 Y-intercept (b) = 59/3 ≈ 19.67

  • Equation 2: y = (3/4)x + 37/4 Slope (m) = 3/4 Y-intercept (b) = 37/4 = 9.25

Step 3: Graph the lines and find intersection

Plot each line on a graph using their slope and y-intercept values, or by finding two points that satisfy each equation. The point where the two lines intersect is the solution to the system of equations. This point appears to be approximately (7,11) based on a visual estimation.

Step 4: Verification

Substitute x = 7 and y = 11 back into both equations to verify:

  • Equation 1: 4(7) + 3(11) = 28 + 33 = 61 (Not exactly 59, due to estimation in graphing)

  • Equation 2: -3(7) + 4(11) = -21 + 44 = 23 (Not exactly 37, due to estimation in graphing)

Since the solution obtained by graphing is an approximation, we see some discrepancies. Let's proceed by calculating the precise solution algebraically to check against the graph's approximate result.

Step 5: Solving algebraically

We'll use the substitution method to solve the system algebraically:

  • Multiply the first equation by 4: 16x + 12y = 236

  • Multiply the second equation by 3: -9x + 12y = 111

  • Subtract the second new equation from the first: 25x = 125

  • Solve for x: x = 5

  • Substitute x = 5 back into either original equation. Using the first: 4(5) + 3y = 59

  • Solve for y: 20 + 3y = 59 => 3y = 39 => y = 13

Final Answer: (5, 13)

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