Question
A system of equations has the solution `(6,\ 2)`.
Here is a graph of one of the equations.
What could the other equation be?
Here is a graph of one of the equations.
What could the other equation be?
Answers
GPT-4o mini
To find another equation that has the solution \( (6, 2) \) along with the given equation, you need to ensure that the point \( (6, 2) \) satisfies both equations. The form of the other equation can vary (linear, quadratic, etc.), but for simplicity, let's create a linear equation.
The general form of a linear equation is:
\[
y = mx + b
\]
Where:
- \( m \) is the slope
- \( b \) is the y-intercept
To create a specific equation, you can choose a slope. For our example, let's say we choose a slope of \( 1 \):
\[
y = x + b
\]
To find the \( b \) value, substitute \( x = 6 \) and \( y = 2 \):
\[
2 = 6 + b
\]
Solving for \( b \):
\[
b = 2 - 6 = -4
\]
Thus, the equation would be:
\[
y = x - 4
\]
You can verify the solution by substituting \( (6, 2) \):
\[
2 = 6 - 4
\]
Since this is true, the equation \( y = x - 4 \) is a valid second equation that, together with the provided one, shares the solution \( (6, 2) \).
You can generate many equations by varying the slope or choosing a different form of the equation, but the key requirement is that both equations must intersect at the point \( (6, 2) \).
The general form of a linear equation is:
\[
y = mx + b
\]
Where:
- \( m \) is the slope
- \( b \) is the y-intercept
To create a specific equation, you can choose a slope. For our example, let's say we choose a slope of \( 1 \):
\[
y = x + b
\]
To find the \( b \) value, substitute \( x = 6 \) and \( y = 2 \):
\[
2 = 6 + b
\]
Solving for \( b \):
\[
b = 2 - 6 = -4
\]
Thus, the equation would be:
\[
y = x - 4
\]
You can verify the solution by substituting \( (6, 2) \):
\[
2 = 6 - 4
\]
Since this is true, the equation \( y = x - 4 \) is a valid second equation that, together with the provided one, shares the solution \( (6, 2) \).
You can generate many equations by varying the slope or choosing a different form of the equation, but the key requirement is that both equations must intersect at the point \( (6, 2) \).