which equation is best represented by this graph
Intercepts and two-point information
The plotted line contains the points (0, 3) and (2, 0). The y-intercept is visible at (0, 3), and the x-intercept is visible at (2, 0). These two points fix a unique linear equation.
Slope from the graph
The slope is the change in \(y\) divided by the change in \(x\) between two points on the line: \[ m=\frac{0-3}{2-0}=-\frac{3}{2}. \] A negative slope matches the downward direction from left to right.
Slope-intercept equation
Slope-intercept form is \(y=m \cdot x + b\). The y-intercept \(b\) equals 3 because the line crosses the y-axis at (0, 3). Substitution gives \[ y=-\frac{3}{2}\cdot x + 3. \]
Best represented equation: \(y=-\frac{3}{2}\cdot x + 3\).
Equivalent equation forms
The same line can be written in several algebraically equivalent ways.
| Form | Equivalent equation | Graph feature emphasized |
|---|---|---|
| Slope-intercept | \(y=-\frac{3}{2}\cdot x + 3\) | Slope and y-intercept |
| Standard | \(3 \cdot x + 2 \cdot y = 6\) | Integer coefficients |
| Intercept | \(\frac{x}{2}+\frac{y}{3}=1\) | x-intercept 2 and y-intercept 3 |
Point substitution consistency
Substitution of (0, 3) yields \(3=-\frac{3}{2}\cdot 0 + 3\), which is true. Substitution of (2, 0) yields \(0=-\frac{3}{2}\cdot 2 + 3 = -3 + 3\), which is true. Agreement at two distinct points confirms the linear equation represented by the graph.
Common pitfalls
Reversing rise and run changes the sign of the slope; the computed slope must match the observed downward trend. Confusing the x-intercept with the y-intercept swaps \(b\) and forces an incorrect line. Fractional slopes are typical when the line drops 3 units while moving 2 units to the right.