To solve the system of linear equations by graphing, we need to rearrange both equations into slope-intercept form (y = mx + b) to find their intersections.
Equation 1: \(2x + 3y = 16.9\)
Let's solve for \(y\):
- Subtract \(2x\) from both sides: \[ 3y = 16.9 - 2x \]
- Divide by 3: \[ y = \frac{16.9 - 2x}{3} \] \[ y = \frac{16.9}{3} - \frac{2}{3}x \] \[ y \approx 5.6333 - 0.6667x \]
Equation 2: \(5x = y + 7.4\)
Let's also solve this for \(y\):
- Subtract \(7.4\) from both sides: \[ y = 5x - 7.4 \]
Now we have the two equations:
- \(y \approx 5.6333 - 0.6667x\)
- \(y = 5x - 7.4\)
Graphing the two equations:
- Equation 1 has a y-intercept of approximately \(5.6333\) and a slope of \(-0.6667\).
- Equation 2 has a y-intercept of \(-7.4\) and a slope of \(5\).
Finding points to graph:
For Equation 1:
- When \(x = 0\): \[ y \approx 5.6333 \]
- When \(x = 5\): \[ y \approx 5.6333 - 0.6667(5) \approx 5.6333 - 3.3335 \approx 2.2998 \]
For Equation 2:
- When \(x = 0\): \[ y = 5(0) - 7.4 = -7.4 \]
- When \(x = 3\): \[ y = 5(3) - 7.4 = 15 - 7.4 = 7.6 \]
Graphing these points:
Plotting the points and drawing the lines for both functions, we can find their intersection point graphically.
Solving Algebraically:
To find the intersection point algebraically:
Set the equations equal to each other:
\[ 5x - 7.4 = 5.6333 - 0.6667x \]
Combine like terms:
\[ 5x + 0.6667x = 5.6333 + 7.4 \] \[ 5.6667x = 13.0333 \]
Now, solve for \(x\):
\[ x \approx \frac{13.0333}{5.6667} \approx 2.303 \] Rounded to the nearest tenth, \(x \approx 2.3\).
Substituting \(x\) back into one of the equations to find \(y\):
Using \(y = 5x - 7.4\):
\[ y \approx 5(2.303) - 7.4 \approx 11.515 - 7.4 \approx 4.115 \] Rounded to the nearest tenth, \(y \approx 4.1\).
Solution:
The solution to the system of linear equations is: \[ (x, y) \approx (2.3, 4.1) \]