Answer :
To solve the system of linear equations by graphing, we need to find the point where the two lines intersect. Let's follow these steps:
1. Rewrite the equations in slope-intercept form (y = mx + b):
- For the first equation \( y + 2.3 = 0.45x \):
- Subtract 2.3 from both sides: \( y = 0.45x - 2.3 \)
- For the second equation \(-2y = 4.2x - 7.8\):
- Divide the entire equation by -2:
- \(-2y = 4.2x - 7.8\)
- \(y = -2.1x + 3.9\)
So, we have the two equations:
[tex]\[ \begin{aligned} y &= 0.45x - 2.3 \quad \text{(Equation 1)} \\ y &= -2.1x + 3.9 \quad \text{(Equation 2)} \end{aligned} \][/tex]
2. Find the intersection point:
- Set the equations equal to each other to find \(x\):
[tex]\[ 0.45x - 2.3 = -2.1x + 3.9 \][/tex]
- Combine like terms:
[tex]\[ 0.45x + 2.1x = 3.9 + 2.3 \][/tex]
- Simplify:
[tex]\[ 2.55x = 6.2 \][/tex]
- Solve for \(x\):
[tex]\[ x = \frac{6.2}{2.55} \approx 2.4 \][/tex]
3. Find the y-coordinate by substituting \(x = 2.4\) back into either equation:
- Using Equation 1:
[tex]\[ y = 0.45(2.4) - 2.3 \][/tex]
[tex]\[ y = 1.08 - 2.3 \][/tex]
[tex]\[ y \approx -1.2 \][/tex]
Thus, the point of intersection is \((2.4, -1.2)\).
So, the solution to the system of linear equations by graphing is [tex]\((2.4, -1.2)\)[/tex].
1. Rewrite the equations in slope-intercept form (y = mx + b):
- For the first equation \( y + 2.3 = 0.45x \):
- Subtract 2.3 from both sides: \( y = 0.45x - 2.3 \)
- For the second equation \(-2y = 4.2x - 7.8\):
- Divide the entire equation by -2:
- \(-2y = 4.2x - 7.8\)
- \(y = -2.1x + 3.9\)
So, we have the two equations:
[tex]\[ \begin{aligned} y &= 0.45x - 2.3 \quad \text{(Equation 1)} \\ y &= -2.1x + 3.9 \quad \text{(Equation 2)} \end{aligned} \][/tex]
2. Find the intersection point:
- Set the equations equal to each other to find \(x\):
[tex]\[ 0.45x - 2.3 = -2.1x + 3.9 \][/tex]
- Combine like terms:
[tex]\[ 0.45x + 2.1x = 3.9 + 2.3 \][/tex]
- Simplify:
[tex]\[ 2.55x = 6.2 \][/tex]
- Solve for \(x\):
[tex]\[ x = \frac{6.2}{2.55} \approx 2.4 \][/tex]
3. Find the y-coordinate by substituting \(x = 2.4\) back into either equation:
- Using Equation 1:
[tex]\[ y = 0.45(2.4) - 2.3 \][/tex]
[tex]\[ y = 1.08 - 2.3 \][/tex]
[tex]\[ y \approx -1.2 \][/tex]
Thus, the point of intersection is \((2.4, -1.2)\).
So, the solution to the system of linear equations by graphing is [tex]\((2.4, -1.2)\)[/tex].
