Which is the best approximation for the solution of the system of equations?

y = A system of equations. y equals negative StartFraction 2 over 5 EndFraction x plus 1. y equals 3 x minus 2.x + 1
y = 3x – 2

A coordinate grid with 2 lines. The first line passes through (0, 1) and (5, negative 2). The second line passes through (0, negative 2) and (1, 1). the lines appear the intersect at a point that is almost to 1 and almost to 1.
(0.45, 0.88)
(0.88, 0.65)
(0.88, 1.1)
(1.3, 0.88)

To find the best approximation for the solution of the given system of equations:

\[ y = -\frac{2}{5}x + 1 \]
\[ y = 3x - 2 \]

we need to solve for the point of intersection of these two lines. 

### Steps to solve the system of equations:
1. Set the equations equal to each other since they both equal \( y \):

\[ -\frac{2}{5}x + 1 = 3x - 2 \]

2. Solve for \( x \):

\[ 1 + 2 = 3x + \frac{2}{5}x \]
\[ 3 = 3x + \frac{2}{5}x \]
\[ 3 = \left(3 + \frac{2}{5}\right)x \]
\[ 3 = \frac{15}{5}x + \frac{2}{5}x \]
\[ 3 = \frac{17}{5}x \]
\[ x = \frac{3 \times 5}{17} \]
\[ x \approx 0.88 \]

3. Substitute \( x \approx 0.88 \) back into either equation to find \( y \). Using \( y = 3x - 2 \):

\[ y = 3(0.88) - 2 \]
\[ y = 2.64 - 2 \]
\[ y \approx 0.64 \]

Now, let's compare the calculated values with the given options:
- (0.45, 0.88)
- (0.88, 0.65)
- (0.88, 1.1)
- (1.3, 0.88)

The closest pair to our calculated values \((0.88, 0.64)\) is \((0.88, 0.65)\).

Therefore, the best approximation for the solution of the system of equations is:

**(0.88, 0.65)**

Would you like further details or have any questions?

Here are 5 questions you might be interested in:

1. How can you graphically solve a system of equations?
2. What are the different methods to solve a system of linear equations?
3. How do you interpret the solution of a system of equations in real-life scenarios?
4. What are some common mistakes to avoid when solving systems of equations?
5. How do you solve a system of equations using substitution and elimination methods?

**Tip:** Always check your solution by substituting the values back into the original equations to verify their accuracy.

Learn how to find the best approximation for the solution of a system of equations using coordinate geometry and linear equations. This problem involves graphing two lines and determining their intersection point. Suitable for high school students studying coordinate geometry and linear equations.

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