Question

m #10
The background of this sketch exhibit is a coordinate grid.Both the x- and y-axis run from negative 6 to 10.The line through the point ( 6 , 2 ) with slope 2 thirds is graphed.
A system of equations has the solution ( 6 , 2 )
(6, 2).



What could the other equation be?

"y" equals 2 thirds "x" minus 1
y=
2
3​x−1

"y" equals negative 3 halves "x" plus 6
y=−
3
2​x+6

"y" equals 1 half "x" minus 1
y=
1
2​x−1

"y" equals 4 "x" minus 2
y=4x−2

Answer 1

To find another equation that has the solution (6, 2) and is different from the line with slope \( \frac{2}{3} \) that goes through (6, 2), you can check which of the provided equations also passes through the point (6, 2).

We'll substitute \( x = 6 \) into each equation and see if \( y = 2 \) results.

1. **For \( y = \frac{2}{3}x - 1 \)**:
\[
y = \frac{2}{3}(6) - 1 = 4 - 1 = 3 \quad (\text{not } (6, 2))
\]

2. **For \( y = -\frac{3}{2}x + 6 \)**:
\[
y = -\frac{3}{2}(6) + 6 = -9 + 6 = -3 \quad (\text{not } (6, 2))
\]

3. **For \( y = \frac{1}{2}x - 1 \)**:
\[
y = \frac{1}{2}(6) - 1 = 3 - 1 = 2 \quad (\text{is } (6, 2))
\]

4. **For \( y = 4x - 2 \)**:
\[
y = 4(6) - 2 = 24 - 2 = 22 \quad (\text{not } (6, 2))
\]

The equation that also has the solution (6, 2) is:
\[
\boxed{y = \frac{1}{2}x - 1}
\]