Match the equations with their graphs.

Matching Equations to Graphs

Let's analyze each equation and its corresponding graph.

Graph 1 (Top Left):

This graph shows a line that passes through approximately (0, 1) and (1, -1). The y-intercept is positive, and the slope is negative. Let's test the equations:

• Equation 1: $$y = -\frac{1}{2}x + 2$$. Y-intercept is 2. (Incorrect for this graph)
• Equation 2: $$y = \frac{1}{2}x - 2$$. Y-intercept is -2. (Incorrect for this graph)
• Equation 3: $$y = -2x - 4$$. Y-intercept is -4. (Incorrect for this graph)
• Equation 4: $$y = 2x + 4$$. Y-intercept is 4. (Incorrect for this graph)

It seems there might be a mismatch or the graph is not precisely plotted for the given options, or the labels are misplaced. However, if we consider the general trend: a steep negative slope and a positive y-intercept, none of the provided options fit perfectly without further clarification or re-examination of the graph's intercepts.

Graph 2 (Top Right):

This graph shows a line that passes through approximately (0, 0) and (1, 2). The y-intercept is 0, and the slope is positive and steep. Let's test the equations:

• Equation 1: $$y = -\frac{1}{2}x + 2$$. Y-intercept is 2. (Incorrect for this graph)
• Equation 2: $$y = \frac{1}{2}x - 2$$. Y-intercept is -2. (Incorrect for this graph)
• Equation 3: $$y = -2x - 4$$. Y-intercept is -4. (Incorrect for this graph)
• Equation 4: $$y = 2x + 4$$. Y-intercept is 4. (Incorrect for this graph)

Again, none of the options perfectly match the visual representation of a line passing through the origin with a steep positive slope. If we look closely, the line appears to pass through (0,0) and rises sharply. If we assume the grid lines are units, the point (1,2) could be on the line. This would suggest y=2x. None of the equations are y=2x. Let's re-evaluate the graphs and equations carefully.

Let's re-examine assuming the labels Б and Г refer to the bottom graphs and the top graphs are unlabeled.

Top Left Graph:

The line passes through approximately (0, 1) and (1, -1). The y-intercept is 1. The slope is $$m = \frac{-1 - 1}{1 - 0} = \frac{-2}{1} = -2$$. This matches the form $$y = mx + b$$. So, $$y = -2x + 1$$. None of the options are $$y = -2x + 1$$. Let's assume the numbers on the axes are correct and the grid is scaled by 1 unit per line.

Let's assume the top left graph is for option 1 or 3.

• Equation 1: $$y = -\frac{1}{2}x + 2$$. Y-intercept is 2.
• Equation 3: $$y = -2x - 4$$. Y-intercept is -4.

The top-left graph appears to have a y-intercept of 1. If we look at the options, none of them have a y-intercept of 1.

Top Right Graph:

The line passes through approximately (0, 0) and (1, 2). The y-intercept is 0. The slope is $$m = \frac{2 - 0}{1 - 0} = 2$$. This would be $$y = 2x$$. None of the options match this exactly.

Let's consider the possibility that the graphs are labeled implicitly. The Russian letters Б and Г are present, which typically correspond to options. Let's assume the top two graphs are for options 1 and 3, and the bottom two graphs (labeled Б and Г) are for options 2 and 4.

Graph for Option Б (Bottom Left):

The line passes through approximately (0, -2) and (1, -1.5). The y-intercept is -2. The slope is $$m = \frac{-1.5 - (-2)}{1 - 0} = \frac{0.5}{1} = 0.5 = \frac{1}{2}$$. This matches the equation $$y = \frac{1}{2}x - 2$$. This corresponds to Option 2.

Graph for Option Г (Bottom Right):

The line passes through approximately (0, 4) and (1, 2). The y-intercept is 4. The slope is $$m = \frac{2 - 4}{1 - 0} = \frac{-2}{1} = -2$$. This matches the equation $$y = -2x + 4$$. None of the options are $$y = -2x + 4$$. Let's re-examine the graph.

Let's try matching based on slope and intercept from the options:

1. $$y = -\frac{1}{2}x + 2$$: Y-intercept = 2, Slope = -0.5 (negative, shallow)
2. $$y = \frac{1}{2}x - 2$$: Y-intercept = -2, Slope = 0.5 (positive, shallow)
3. $$y = -2x - 4$$: Y-intercept = -4, Slope = -2 (negative, steep)
4. $$y = 2x + 4$$: Y-intercept = 4, Slope = 2 (positive, steep)

Let's re-evaluate the graphs with these properties:

• Top Left Graph: Y-intercept appears to be 1 or 2. Slope is negative and steep. This could potentially match $$y = -2x + ext{something}$$. If the y-intercept is 2, it would match equation 1 if the slope was -1/2, not steep. If the slope is -2, it's steep and negative. None of the options have y-intercept 1 or 2 with a steep negative slope. However, if the y-intercept is 2, and the slope is -2, the line would pass through (0,2) and (1,0). This is close to what is shown, but not exactly. If we assume the y-intercept is 2, it could be option 1, but the slope is not -1/2, it's steeper. Let's consider the possibility of mislabeling or imprecise drawing. If we assume the y-intercept is 2 and the slope is -1/2, the line should pass through (0,2) and (2,1). This doesn't match.
• Top Right Graph: Y-intercept appears to be 0. Slope is positive and steep. This would best fit $$y=2x$$. Option 4 has a positive steep slope, but y-intercept 4. If we assume the y-intercept is 4, the line would pass through (0,4) and (1,6). This does not match. If we assume it passes through (0,0) and (1,2), it's $$y=2x$$.
• Bottom Left Graph (Б): Y-intercept appears to be -2. Slope is positive and shallow. This perfectly matches Option 2: $$y = \frac{1}{2}x - 2$$.
• Bottom Right Graph (Г): Y-intercept appears to be 4. Slope is negative and steep. This matches Option 4: $$y = 2x + 4$$, if we swap the slope sign and y-intercept. Let's re-examine. The line passes through approximately (0,4) and (-2,0). The slope is $$m = \frac{0-4}{-2-0} = \frac{-4}{-2} = 2$$. This is a positive slope. This fits the description of $$y=2x+4$$. So, the bottom right graph (Г) corresponds to Option 4.

Let's re-evaluate the top graphs based on the remaining options (1 and 3).

• Top Left Graph: Y-intercept appears to be 2. Slope is negative and steep. This best fits Option 3: $$y = -2x - 4$$, if the y-intercept were -4, or if the line were $$y=-2x+2$$. If we consider the possibility that the top left graph is for Option 1 ($$y = -\frac{1}{2}x + 2$$), then the y-intercept is 2 and the slope is -1/2 (negative, shallow). This is somewhat consistent with the top left graph if we adjust our reading of the steepness and intercept. Let's assume the y-intercept is 2. The line passes through (0,2). For option 1, it would also pass through (2,1) and (4,0). This doesn't quite match the steepness. For option 3, $$y=-2x-4$$, y-intercept is -4.

There seems to be a significant discrepancy between the drawn graphs and the provided equations, or the labeling is unconventional.

Let's assume the top graphs correspond to the first two equations and the bottom graphs to the last two.

Hypothesis 1:

• Top Left Graph: Equation 1 ($$y = -\frac{1}{2}x + 2$$) - Y-intercept 2, negative shallow slope.
• Top Right Graph: Equation 2 ($$y = \frac{1}{2}x - 2$$) - Y-intercept -2, positive shallow slope.
• Bottom Left Graph (Б): Equation 3 ($$y = -2x - 4$$) - Y-intercept -4, negative steep slope.
• Bottom Right Graph (Г): Equation 4 ($$y = 2x + 4$$) - Y-intercept 4, positive steep slope.

This hypothesis does not align well with the visuals.

Hypothesis 2 (Based on visual characteristics):

• Option 2 ($$y = \frac{1}{2}x - 2$$): Y-intercept = -2, Slope = 0.5 (positive, shallow). This matches Bottom Left Graph (Б).
• Option 4 ($$y = 2x + 4$$): Y-intercept = 4, Slope = 2 (positive, steep). This matches Bottom Right Graph (Г).

Now let's look at the remaining graphs and options.

• Top Left Graph: Appears to have a positive y-intercept and a steep negative slope. Let's assume the y-intercept is 2. If the slope is -2, it would be $$y=-2x+2$$. Option 3 is $$y=-2x-4$$. The slope is steep negative, but the y-intercept is -4. The graph seems to have a positive y-intercept. If we assume the y-intercept is 2, then it might be intended for option 1, but the slope in option 1 is -1/2 (shallow).
• Top Right Graph: Appears to have a positive y-intercept and a steep positive slope. Let's assume the y-intercept is 4. If the slope is 2, it would be $$y=2x+4$$. This is option 4, which we've already matched. This suggests an error in our assignments or the image.

Let's strictly match based on visually identifiable features, prioritizing y-intercept and then slope steepness and sign.

Revised Matching:

1. Equation 1: $$y = -\frac{1}{2}x + 2$$
   • Y-intercept: 2 (positive)
   • Slope: -0.5 (negative, shallow)
   • Likely Graph: Top Left Graph (Y-intercept around 1-2, negative slope). The steepness might be drawn inaccurately.
2. Equation 2: $$y = \frac{1}{2}x - 2$$
   • Y-intercept: -2 (negative)
   • Slope: 0.5 (positive, shallow)
   • Likely Graph: Bottom Left Graph (Б) (Y-intercept -2, positive shallow slope). This is a strong match.
3. Equation 3: $$y = -2x - 4$$
   • Y-intercept: -4 (negative)
   • Slope: -2 (negative, steep)
   • Likely Graph: None of the graphs clearly shows a y-intercept of -4 with a steep negative slope. The top left graph has a negative slope, but the intercept is not -4.
4. Equation 4: $$y = 2x + 4$$
   • Y-intercept: 4 (positive)
   • Slope: 2 (positive, steep)
   • Likely Graph: Bottom Right Graph (Г) (Y-intercept around 4, positive steep slope). This is a strong match.

Given the strong matches for options 2 and 4, let's re-examine the remaining graphs for options 1 and 3.

Revisiting Top Left Graph:

It shows a negative slope. The y-intercept is positive, around 1 or 2. The slope appears steeper than -0.5. If we consider option 1 ($$y = -0.5x + 2$$), the y-intercept is 2, slope is -0.5. If we consider option 3 ($$y = -2x - 4$$), the y-intercept is -4, slope is -2. The top-left graph's intercept is positive, so it cannot be option 3. If it's option 1, the slope appears steeper than -0.5. However, visual estimation can be misleading.

Revisiting Top Right Graph:

It shows a positive slope. The y-intercept appears to be 0 or slightly positive. The slope is steep. Option 4 ($$y=2x+4$$) has a steep positive slope but a y-intercept of 4. This graph does not seem to match y=2x+4.

Conclusion based on the most likely matches:

• Graph Б (Bottom Left) matches Equation 2: $$y = \frac{1}{2}x - 2$$.
• Graph Г (Bottom Right) matches Equation 4: $$y = 2x + 4$$.

Now we need to assign Equations 1 and 3 to the top two graphs.

• Top Left Graph: Negative slope, positive y-intercept (around 1 or 2). This is more consistent with Equation 1 ($$y = -\frac{1}{2}x + 2$$) than Equation 3 ($$y = -2x - 4$$), which has a negative y-intercept. The steepness of the top-left graph appears more like -2 than -0.5, which is a contradiction. Assuming the y-intercept is the primary identifier, it fits better with Equation 1 if the y-intercept is indeed 2.
• Top Right Graph: Positive slope, y-intercept around 0. Option 3 ($$y = -2x - 4$$) has a negative slope and negative intercept. Option 1 ($$y = -\frac{1}{2}x + 2$$) has a negative slope and positive intercept. This graph has a positive slope. This reinforces the idea that the top right graph might correspond to an equation not listed, or there's a severe misrepresentation.

Let's assume the labels Б and Г apply to the bottom graphs, and the top two are implicitly assigned based on visual cues, even if imperfect.

Final Assignment based on best visual fit:

• Equation 1: $$y = -\frac{1}{2}x + 2$$ matches the Top Left Graph (despite potential inaccuracy in slope steepness).
• Equation 2: $$y = \frac{1}{2}x - 2$$ matches the Bottom Left Graph (Б).
• Equation 3: $$y = -2x - 4$$ is problematic. None of the graphs perfectly fit. The top left graph has a negative slope, but the intercept is positive. If we had to choose, and assuming the top left graph is NOT option 1, this would be the only other option with a negative slope. But the intercept is wrong.
• Equation 4: $$y = 2x + 4$$ matches the Bottom Right Graph (Г).

Given the ambiguity and potential inaccuracies in the drawing, the most confident matches are for equations 2 and 4 with graphs Б and Г respectively.

Let's proceed with the most plausible assignments:

• Graph 1 (Top Left) → Equation 1: $$y = -\frac{1}{2}x + 2$$ (Y-intercept approx 2, negative slope)
• Graph 2 (Top Right) → This graph does not clearly match any remaining equation (3). It shows a positive slope and a y-intercept near 0. Equation 3 has a negative slope and y-intercept of -4. There might be an error in the question or diagram.
• Graph 3 (Bottom Left, Б) → Equation 2: $$y = \frac{1}{2}x - 2$$ (Y-intercept -2, positive shallow slope)
• Graph 4 (Bottom Right, Г) → Equation 4: $$y = 2x + 4$$ (Y-intercept 4, positive steep slope)

Due to the discrepancies, it's impossible to provide a perfect one-to-one mapping for all. However, based on the strongest visual cues:

• Graph Б corresponds to 2) $$y = \frac{1}{2}x - 2$$
• Graph Г corresponds to 4) $$y = 2x + 4$$

For the top two graphs:

• The Top Left Graph (negative slope, positive intercept around 1-2) is most plausibly associated with 1) $$y = -\frac{1}{2}x + 2$$.
• The Top Right Graph (positive slope, intercept around 0) does not clearly match 3) $$y = -2x - 4$$ (negative slope, intercept -4). It might be misdrawn or intended to match a different equation. If forced to match, and assuming a significant drawing error, it's difficult to assign.

Final most probable matching:

• 1) $$y = -\frac{1}{2}x + 2$$ – Top Left Graph
• 2) $$y = \frac{1}{2}x - 2$$ – Graph Б
• 3) $$y = -2x - 4$$ – No clear match.
• 4) $$y = 2x + 4$$ – Graph Г