To match each piece of information with the appropriate equation, let's analyze each scenario step-by-step and identify the corresponding equation.
1. For [tex]\( m=1, b=-3 \)[/tex]:
- This describes a line in slope-intercept form: [tex]\( y = mx + b \)[/tex].
- Given [tex]\( m = 1 \)[/tex] and [tex]\( b = -3 \)[/tex], the equation becomes:
[tex]\[
y = 1x - 3 \quad \text{or simply} \quad y = x - 3.
\][/tex]
- To convert this to standard form [tex]\( Ax + By = C \)[/tex],
[tex]\[
y = x - 3 \quad \implies \quad x - y = 3.
\][/tex]
- Therefore, the corresponding equation is [tex]\( x - y = 3 \)[/tex].
2. For [tex]\( m=1, (-1,2) \)[/tex]:
- This describes a line with a given slope [tex]\( m = 1 \)[/tex] passing through the point [tex]\((-1,2)\)[/tex].
- We use the point-slope form: [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Substituting [tex]\( m = 1 \)[/tex], [tex]\( x_1 = -1 \)[/tex], and [tex]\( y_1 = 2 \)[/tex]:
[tex]\[
y - 2 = 1(x + 1) \quad \implies \quad y - 2 = x + 1 \quad \implies \quad x - y = -1.
\][/tex]
- Therefore, the corresponding equation is [tex]\( x - y = -1 \)[/tex].
3. For [tex]\((-2,3), (-3,4)\)[/tex]:
- This describes a line passing through two points: [tex]\((-2,3)\)[/tex] and [tex]\((-3,4)\)[/tex].
- First, we calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 3}{-3 - (-2)} = \frac{1}{-1} = -1.
\][/tex]
- Using the point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex] with [tex]\((-2,3)\)[/tex]:
[tex]\[
y - 3 = -1(x + 2) \quad \implies \quad y - 3 = -x - 2 \quad \implies \quad x + y = 1.
\][/tex]
- Therefore, the corresponding equation is [tex]\( x + y = 1 \)[/tex].
In summary, the matches are:
- [tex]\( m=1, b=-3 \)[/tex] matches with [tex]\( x - y = 3 \)[/tex].
- [tex]\( m=1, (-1,2) \)[/tex] matches with [tex]\( x - y = -1 \)[/tex].
- [tex]\((-2,3), (-3,4) \)[/tex] matches with [tex]\( x + y = 1 \)[/tex].
