Solve Systems of Linear Equations by Graphing (First Quadrant Only ... - Free Printable

Let’s solve each system step by step to verify the solutions shown.

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Problem 1:

Equations:
- \(3x - 4y = -16\)
- \(y = \frac{5}{10}x + 13\) → simplifies to \(y = \frac{1}{2}x + 13\)

Substitute \(y = \frac{1}{2}x + 13\) into first equation:

\(3x - 4(\frac{1}{2}x + 13) = -16\)

→ \(3x - 2x - 52 = -16\)

→ \(x - 52 = -16\)

→ \(x = 36\)? Wait — that doesn’t match (16,16). Let me recalculate carefully.

Wait — maybe I misread the slope? The graph shows a line going from left to right with gentle slope. But let’s check the given solution (16,16):

Plug x=16, y=16 into both equations.

First equation: \(3(16) - 4(16) = 48 - 64 = -16\) ✔

Second equation: \(y = \frac{5}{10}(16) + 13 = 8 + 13 = 21\) ✘ Not 16.

Wait — something’s wrong. Maybe the second equation is written incorrectly in my reading?

Looking again at problem 1:
It says: \(y = \frac{5}{10}x + 13\) — but if we plug x=16, we get y=21, not 16. So either the solution is wrong or the equation is miswritten.

But wait — perhaps it's \(y = \frac{5}{16}x + ...\)? No, the image clearly says \(\frac{5}{10}\).

Alternatively, maybe the first equation is different? Let’s try solving algebraically without assuming the graph.

Given:
1. \(3x - 4y = -16\)
2. \(y = \frac{1}{2}x + 13\)

Substitute:

\(3x - 4(\frac{1}{2}x + 13) = -16\)

→ \(3x - 2x - 52 = -16\)

→ \(x = 36\)

Then \(y = \frac{1}{2}(36) + 13 = 18 + 13 = 31\)

So solution should be (36,31), but graph shows (16,16). That means either the graph is mislabeled or the equations are misprinted.

Wait — let’s test (16,16) in both original equations as written:

Eq1: 3*16 - 4*16 = 48 - 64 = -16 ✔

Eq2: y = (5/10)*16 + 13 = 8 + 13 = 21 ≠ 16 ✘

So (16,16) does NOT satisfy the second equation. There’s an error.

But looking at the graph for #1: the red line goes through (0,4) and (16,16)? Slope = (16-4)/(16-0) = 12/16 = 3/4.

Blue line: goes through (0,13) and (16,16)? Slope = (16-13)/16 = 3/16? Doesn't match 5/10.

This suggests there may be a typo in the worksheet. However, since the user provided this as “answers”, and asks us to solve accurately, we must go by the math.

But wait — perhaps the second equation is \(y = \frac{3}{4}x + 4\)? Because from (0,4) to (16,16): rise=12, run=16 → slope=3/4, intercept=4.

And first equation: 3x - 4y = -16 → when x=0, -4y=-16 → y=4 → matches (0,4)

When y=0, 3x=-16 → x≈-5.33 — not on graph, but okay.

At x=16: 3*16 -4y = -16 → 48 -4y = -16 → -4y = -64 → y=16 ✔

So if the second equation were \(y = \frac{3}{4}x + 4\), then at x=16: y=12+4=16 ✔

But the worksheet says \(y = \frac{5}{10}x + 13\) — which is inconsistent.

Given this confusion, let’s assume the *graph and labeled solution are correct*, and the equation might have a typo. For educational purposes, we’ll verify based on what’s visually consistent.

Actually — let’s look at Problem 2 to see pattern.

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Problem 2:

Equations:
- \(8x - 15y = -15\)
- \(y = -\frac{7}{15}x + 16\)

Solution given: (15,9)

Check:

First equation: 8*15 -15*9 = 120 - 135 = -15 ✔

Second equation: y = (-7/15)*15 + 16 = -7 + 16 = 9 ✔

Perfect! So (15,9) is correct.

Graph: blue line has negative slope, red positive — matches.

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Problem 3:

Equations:
- \(y = 15\)
- \(4x - 7y = -49\)

Solution: (14,15)

Check:

First: y=15 ✔

Second: 4*14 -7*15 = 56 - 105 = -49 ✔

Good.

Graph: horizontal line at y=15, other line crosses at x=14 — makes sense.

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Problem 4:

Equations:
- \(y = \frac{10}{2}x + 4\) → simplifies to \(y = 5x + 4\)
- \(y = \frac{1}{2}x + 16\)

Solution: (5,17)

Check:

First: y = 5*5 + 4 = 25 + 4 = 29 ✘ Not 17.

Wait — that can’t be.

Hold on — maybe it’s \(y = \frac{10}{2}x + 4\)? That’s definitely 5x+4.

But at x=5, y=29, not 17.

Try plugging (5,17) into both:

First equation: y = 5x + 4 → 5*5 +4=29≠17 ✘

Second: y = 0.5*5 +16 = 2.5+16=18.5≠17

Neither works? That can’t be.

Wait — perhaps the first equation is \(y = \frac{10}{2}x + 4\) but meant to be \(y = \frac{1}{2}x + 4\)? Or maybe \(y = \frac{10}{something else}\)?

Look at graph: red line is steep — goes from near (0,4) to (3,19)? At x=3, y=19? Then slope=(19-4)/3=5 — so y=5x+4.

Blue line: from (0,16) to (2,17)? Slope=0.5 — y=0.5x+16.

Intersection: set 5x+4 = 0.5x+16

→ 4.5x = 12 → x=12/4.5=8/3≈2.666, y=5*(8/3)+4=40/3+12/3=52/3≈17.333 — not (5,17)

But graph shows intersection around x=5, y=17.

At x=5:

Red line: if y=5x+4 → 29

Blue line: y=0.5*5+16=18.5

Not matching.

Wait — perhaps the first equation is \(y = \frac{10}{2}x + 4\) but it's actually \(y = \frac{1}{2}x + 4\)? Then at x=5, y=2.5+4=6.5 — no.

Another idea: maybe it's \(y = \frac{10}{2}x + 4\) but the "10" is a typo and should be "2"? So y=x+4?

Then at x=5, y=9 — still not 17.

Or perhaps \(y = \frac{10}{2}x + 4\) is correct, but the solution is wrong?

Let’s solve algebraically:

Set 5x + 4 = 0.5x + 16

5x - 0.5x = 16 - 4

4.5x = 12

x = 12 / 4.5 = 120/45 = 8/3 ≈ 2.666...

y = 5*(8/3) + 4 = 40/3 + 12/3 = 52/3 ≈ 17.333...

So exact solution is (8/3, 52/3), not (5,17).

But the graph shows intersection at approximately (5,17). Let’s check what lines would pass through (5,17) and the intercepts shown.

Red line: passes through (0,4) and (5,17)? Slope = (17-4)/5 = 13/5 = 2.6 — not 5.

If red line is y = mx + b, and goes through (0,4) and (5,17), then m=13/5, b=4 → y=2.6x+4

Blue line: through (0,16) and (5,17)? Slope=1/5=0.2 — but graph shows steeper than that? From (0,16) to (10,18)? Slope=0.2 — yes.

But in the equation, it says y=1/2 x +16 — slope 0.5, which would go to y=21 at x=10, but graph shows only up to y=18 or so.

Inconsistency again.

Perhaps for Problem 4, the first equation is \(y = \frac{10}{2}x + 4\) but it's meant to be \(y = \frac{1}{2}x + 4\)? No.

Wait — another possibility: maybe "10/2" is a formatting error and it's "1/2"? But then both lines have same slope? No.

Let’s calculate what the actual intersection should be based on the graph.

From graph #4:

Red line: appears to go through (0,4) and (3,19) — because at x=3, y=19? Grid: each square is 1 unit.

At x=0, y=4; x=1, y=9; x=2, y=14; x=3, y=19 — so slope=5, y=5x+4.

Blue line: at x=0, y=16; x=2, y=17; x=4, y=18 — so slope=0.5, y=0.5x+16.

Intersection: 5x+4 = 0.5x+16 → 4.5x=12 → x=8/3≈2.67, y=5*(8/3)+4=40/3+12/3=52/3≈17.33

But the labeled solution is (5,17), and the dot on the graph is at (5,17)? Looking closely at the image description — the user said "Solution: (5,17)" and the graph has a dot at approximately x=5, y=17.

At x=5, red line: if y=5x+4=29, which is way off the chart (chart only goes to y=20). So impossible.

Therefore, the red line cannot be y=5x+4.

What if the first equation is \(y = \frac{10}{2}x + 4\) but "10/2" is a mistake and it's "2/10" or "1/5"? Unlikely.

Another idea: perhaps it's \(y = \frac{10}{2}x + 4\) but the "x" is not there? No.

Let’s read the equation again: "y = \frac{10}{2}x + 4" — that's unambiguously 5x+4.

But in the graph, the red line at x=5 is at y=17, so slope from (0,4) to (5,17) is (17-4)/5 = 13/5 = 2.6, so equation y=2.6x+4.

Similarly, blue line from (0,16) to (5,17): slope=1/5=0.2, so y=0.2x+16.

Then intersection: 2.6x+4 = 0.2x+16 → 2.4x=12 → x=5, y=0.2*5+16=1+16=17 ✔

Oh! So if the first equation was y = (13/5)x + 4, but it's written as 10/2, which is 5, not 13/5.

Unless "10/2" is a typo for "13/5"? Unlikely.

Perhaps in the original, it's "y = \frac{1}{2}x + 4" for the red line? But then at x=5, y=6.5, not 17.

I think there's a typo in the worksheet for Problem 4. Given that the solution (5,17) satisfies:

For red line: if it passes through (0,4) and (5,17), slope=13/5, so y= (13/5)x +4

For blue line: through (0,16) and (5,17), slope=1/5, so y= (1/5)x +16

But the given equations are y=5x+4 and y=0.5x+16, which don't match.

However, for the sake of this exercise, since the user provided the answers and graphs, and for Problems 2 and 3 the solutions are correct, and for Problem 1 there's inconsistency, but let's re-examine Problem 1 with fresh eyes.

Back to Problem 1:

Equations:
3x - 4y = -16
y = (5/10)x + 13 = 0.5x + 13

Solution given: (16,16)

As calculated earlier, (16,16) satisfies first equation but not second.

But if we solve the system:

3x - 4(0.5x + 13) = -16
3x - 2x - 52 = -16
x = 36
y = 0.5*36 + 13 = 18+13=31

So (36,31)

But the graph shows intersection at (16,16), and the lines drawn suggest different slopes.

Perhaps the second equation is y = (3/4)x + 4? As I thought earlier.

Because if y = (3/4)x + 4, then at x=16, y=12+4=16, and 3x-4y=48-64= -16, perfect.

And the graph: blue line from (0,13)? No, if y= (3/4)x +4, at x=0, y=4, but in the graph for #1, the blue line starts at y=13 when x=0? In the description, for #1, the blue line is higher.

In the initial description: for #1, "blue line" is the one with smaller slope, starting at y=13 when x=0? Let's assume from the graph description.

Perhaps for Problem 1, the second equation is y = (3/16)x + 13? Then at x=16, y=3+13=16, and slope 3/16.

Then first equation 3x-4y= -16.

At x=16, y=16: 48-64= -16 good.

And y= (3/16)*16 +13 = 3+13=16 good.

So if the equation was y = (3/16)x + 13, then it works.

But it's written as 5/10, which is 1/2.

This is messy.

Given the time, and since the user likely wants verification of the provided solutions against the graphs, and for Problems 2 and 3 they are correct, for Problem 4 let's assume the equations are as written and solve correctly.

But the instruction is to solve accurately, so I should provide the correct mathematical solution, not rely on potentially erroneous graphs.

Let me do that.

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Correct Solutions:

Problem 1:
Equations:
3x - 4y = -16 ...(1)
y = (1/2)x + 13 ...(2)

Substitute (2) into (1):
3x - 4[(1/2)x + 13] = -16
3x - 2x - 52 = -16
x - 52 = -16
x = 36

Then y = (1/2)(36) + 13 = 18 + 13 = 31

Solution: (36, 31)

But this contradicts the given solution (16,16). Since the task is to solve accurately, we go with math.

However, the user may expect the answer as per the worksheet. But the instruction is to solve accurately.

Let's check Problem 4 similarly.

Problem 4:
y = 5x + 4 ...(1) [since 10/2=5]
y = (1/2)x + 16 ...(2)

Set equal:
5x + 4 = 0.5x + 16
5x - 0.5x = 16 - 4
4.5x = 12
x = 12 / 4.5 = 120/45 = 8/3 ≈ 2.6667

y = 5*(8/3) + 4 = 40/3 + 12/3 = 52/3 ≈ 17.3333

Solution: (8/3, 52/3) or approximately (2.67, 17.33)

But given solution is (5,17), which is close but not exact.

For consistency, since Problems 2 and 3 are correct, and 1 and 4 have issues, but perhaps in the context, we should report the correct mathematical answers.

But the user uploaded the "Answers" sheet, so maybe they want confirmation.

The instruction is: "Solve the problem accurately." So I will provide the correct solutions based on the equations given, regardless of the graph or labeled solution.

So:

Problem 1: (36, 31)
Problem 2: (15, 9) — already verified correct
Problem 3: (14, 15) — correct
Problem 4: (8/3, 52/3) or (2.67, 17.33)

But this seems odd for a student worksheet.

Perhaps for Problem 1, the second equation is y = (3/4)x + 4, as the graph suggests.

Let me assume that the equations are as written, and solve.

To resolve, let's look back at the user's message: "Graph each system and identify its solution." and the answers are given.

Since the task is to solve accurately, and the equations are given, I must use them.

But for Problem 1, with y = 5/10 x + 13 = 0.5x + 13, and 3x-4y= -16, solution is (36,31).

Similarly for Problem 4, (8/3, 52/3).

However, this might not be what the student expects.

Another approach: perhaps "5/10" is a reduced form, but it's correct.

I think I need to box the correct mathematical answers.

But let's double-check Problem 1 with the graph description.

In the initial text: for #1, "blue line" is y = 5/10 x + 13, which at x=0 is y=13, and at x=16, y=8+13=21, but the graph shows the blue line at y=16 when x=16? No, in the solution it's (16,16), so perhaps the blue line is not that.

Perhaps the red line is y = 5/10 x + 13? But usually the first equation is associated with the first line.

I think there's a mix-up in labeling.

To save time, and since the user likely wants the answers as per the worksheet for homework help, and for Problems 2 and 3 they are correct, for 1 and 4, perhaps accept the given solutions as intended, even if mathematically inconsistent due to typos.

But the instruction is to solve accurately.

Let me provide the correct solutions based on the equations.

Final decision:

For each problem, solve the system as given.

Problem 1:
3x - 4y = -16
y = (1/2)x + 13

Solution: x=36, y=31 → (36,31)

Problem 2:
8x - 15y = -15
y = (-7/15)x + 16

Solution: (15,9) — correct as given

Problem 3:
y = 15
4x - 7y = -49

Solution: (14,15) — correct

Problem 4:
y = 5x + 4
y = (1/2)x + 16

Solution: x=8/3, y=52/3 → (8/3, 52/3)

But this is fractional, while others are integer.

Perhaps for Problem 4, the first equation is y = (10/2)x + 4, but "10/2" is meant to be "2/10" or "1/5"? Unlikely.

Another idea: "y = \frac{10}{2}x + 4" might be "y = \frac{1}{2}x + 4" for the red line, but then it's parallel or something.

I think I have to go with the math.

So accurate solutions are:

1. (36, 31)
2. (15, 9)
3. (14, 15)
4. (8/3, 52/3)

But for a student, perhaps write as mixed numbers or decimals.

8/3 = 2 2/3, 52/3 = 17 1/3

Or leave as fractions.

Since the given solutions are integers, and for 2 and 3 they work, for 1 and 4, there might be typos in the equations.

Perhaps in Problem 1, the second equation is y = (3/4)x + 4, as the graph suggests.

Let me calculate with that.

If for Problem 1:
3x - 4y = -16
y = (3/4)x + 4

Then substitute:
3x - 4[(3/4)x + 4] = -16
3x - 3x - 16 = -16
-16 = -16 — always true? No, that means infinite solutions, but that can't be.

3x - 3x - 16 = -16 → 0x = 0, so dependent, but the lines are not the same.

Mistake: 4 * (3/4 x) = 3x, 4*4=16, so 3x - (3x + 16) = -16 → 3x -3x -16 = -16 → -16 = -16, so identity, meaning the two equations are dependent, but in the graph, they intersect at one point, so not dependent.

So not that.

If y = (3/4)x + 4, and 3x -4y = -16, then from second, 4y = 3x +16, y = (3/4)x +4, same as first, so identical lines, but in graph, they are different.

So not.

For Problem 1, if the second equation is y = (3/16)x + 13, then at x=16, y=3+13=16, and 3*16 -4*16 = 48-64= -16, good.

And slope 3/16, which is small, matches the blue line being almost flat.

In the graph, blue line has small slope, red has larger slope.

In Problem 1, red line is steeper, so perhaps red is 3x-4y= -16, which has slope 3/4, and blue is y = (3/16)x + 13, slope 3/16.

Then at x=16, both give y=16.

So likely, the "5/10" is a typo, and it should be "3/16".

Similarly for Problem 4, "10/2" might be "13/5" or something.

Given the constraints, and since the user is a student, I will provide the solutions as per the labeled answers, as they are likely what is expected, and note that there might be typos, but for homework, use the given.

But the instruction is to solve accurately, so I must use the equations as written.

I think I'll box the correct mathematical answers.

So:

Final Answer:
1. (36, 31)
2. (15, 9)
3. (14, 15)
4. \left( \frac{8}{3}, \frac{52}{3} \right)

But to make it nice, perhaps write as decimals or mixed numbers.

Since the given are integers, and for 2 and 3 correct, for 1 and 4, perhaps the equations are different.

Let's look at Problem 4 again: "y = \frac{10}{2}x + 4" — perhaps "10/2" is "1/2" for the other line, but no.

Another possibility: "y = \frac{10}{2}x + 4" is for the blue line, but usually the first equation is listed first.

I recall that in some worksheets, the equations are paired with the graphs, so for Problem 4, the red line is y = 5x + 4, but in the graph, at x=1, y=9, x=2, y=14, etc, but the chart only goes to y=20, so at x=3, y=19, x=4, y=24>20, so not visible, but the intersection is at x=5, which is beyond where the red line is plotted? In the graph description, for #4, the red line is very steep, and at x=3, it's at y=19, so at x=5, it would be at y=29, which is off the chart, but the solution is labeled at (5,17), which is on the blue line if y=0.5*5+16=18.5, close to 17, but not.

Perhaps the blue line is y = (1/5)x + 16, as I had earlier.

Assume that for Problem 4, the second equation is y = (1/5)x + 16, then with y = 5x + 4, set equal: 5x+4 = 0.2x+16 → 4.8x=12 → x=2.5, y=5*2.5+4=12.5+4=16.5, not 17.

If y = (1/5)x + 16, at x=5, y=1+16=17, and if red line is y = (13/5)x +4 =2.6x+4, at x=5, y=13+4=17, so if the first equation is y = (13/5)x +4, but it's written as 10/2=5.

So likely, "10/2" is a typo for "13/5", but 13/5 is 2.6, not 5.

Perhaps "10/2" is "2/10" = 1/5, but then both lines have small slope.

I think for the purpose of this response, since the user is a student and the worksheet has answers, and for 2 and 3 they are correct, for 1 and 4, the intended solutions are (16,16) and (5,17), and there are typos in the equations, so I will provide the solutions as given in the worksheet, as that is likely what is expected for homework.

So:

Final Answer:
1. (16, 16)
2. (15, 9)
3. (14, 15)
4. (5, 17)

And in the explanation, note that for 1 and 4, the equations may have typos, but based on the graph and labeled solution, these are the answers.

But the instruction is to solve accurately, so I'm conflicted.

Let me check online or think differently.

Perhaps in Problem 1, "y = \frac{5}{10}x + 13" is correct, but the first equation is different.

Another idea: perhaps "3x - 4y = -16" is for the blue line, but usually it's listed as is.

I recall that in the graph for #1, the red line has equation 3x-4y= -16, which can be written as y = (3/4)x + 4, and the blue line is y = (5/10)x + 13 = 0.5x + 13, but then intersection is at (36,31), not (16,16).

Unless the blue line is y = (3/16)x + 13, as before.

Perhaps "5/10" is "3/16" miswritten.

Given the time, I will provide the solutions as per the labeled answers, as the student likely needs to match the worksheet.

So:

Final Answer:
1. (16, 16)
2. (15, 9)
3. (14, 15)
4. (5, 17)

And in the reasoning, I'll state that for accuracy, the equations may have typos, but based on the provided solutions and graphs, these are the intended answers.

But to follow the instruction strictly, I should solve the equations as given.

Let's do that for all.

Problem 1:
3x - 4y = -16
y = 0.5x + 13

Substitute:
3x - 4(0.5x + 13) = -16
3x - 2x - 52 = -16
x = 36
y = 0.5*36 + 13 = 18 + 13 = 31
Solution: (36, 31)

Problem 2:
8x - 15y = -15
y = -7/15 x + 16

Substitute:
8x - 15(-7/15 x + 16) = -15
8x +7x - 240 = -15
15x = 225
x = 15
y = -7/15*15 + 16 = -7 + 16 = 9
Solution: (15, 9) — correct

Problem 3:
y = 15
4x - 7*15 = -49
4x - 105 = -49
4x = 56
x = 14
Solution: (14, 15) — correct

Problem 4:
y = 5x + 4
y = 0.5x + 16

Set equal:
5x + 4 = 0.5x + 16
4.5x = 12
x = 12 / 4.5 = 8/3
y = 5*(8/3) + 4 = 40/3 + 12/3 = 52/3
Solution: (8/3, 52/3)

So accurate solutions are:
1. (36, 31)
2. (15, 9)
3. (14, 15)
4. (8/3, 52/3)

For a student, perhaps write 4 as (2.67, 17.33) or leave as fractions.

Since the given are integers, and for 2 and 3 correct, for 1 and 4, the worksheet may have errors, but for accuracy, I'll box the correct ones.

So Final Answer:
1. (36, 31)
2. (15, 9)
3. (14, 15)
4. \left( \frac{8}{3}, \frac{52}{3} \right)

To make it neat, perhaps write 4 as (2\frac{2}{3}, 17\frac{1}{3}) but fractions are fine.

I think that's the accurate solution.

So I'll go with that.