Graph the system of linear equations.

To graph the system of linear equations given by:

1. \(-\frac{1}{2}y = \frac{1}{2}x + 5\)
2. \(y = 2x + 2\)

We will first rewrite both equations in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Rewriting Equation 1

Starting with \(-\frac{1}{2}y = \frac{1}{2}x + 5\), we multiply both sides by \(-2\) to isolate \(y\):

\[ y = -x - 10 \]

Equation 2

The second equation is already in slope-intercept form:

\[ y = 2x + 2 \]

Finding the Intersection (Solution)

To find the solution of the system, we need to set the two equations equal to each other:

\[ -x - 10 = 2x + 2 \]

Solving for \(x\)

Let's solve for \(x\):

1. Add \(x\) to both sides: \[ -10 = 3x + 2 \]

2. Subtract \(2\) from both sides: \[ -12 = 3x \]

3. Divide by \(3\): \[ x = -4 \]

Finding \(y\)

Now plug \(x = -4\) back into either original equation to find \(y\). Using the second equation:

\[ y = 2(-4) + 2 = -8 + 2 = -6 \]

Solution

The solution to the system of equations is:

\[ (x, y) = (-4, -6) \]

Graphing the Equations

1. For \(y = -x - 10\):

   • Y-intercept: at (0, -10)
   • Slope: -1 (down 1, right 1)

2. For \(y = 2x + 2\):

   • Y-intercept: at (0, 2)
   • Slope: 2 (up 2, right 1)

Conclusion

The graphical representation of both lines will intersect at the point \((-4, -6)\), which is the solution to the system.

So the final answer is: \[ \text{The solution to the system is } (-4, -6). \]

Consider the system of linear equations.

2y = x + 10

3y = 3x + 15

Which statements about the system are true? Check all that apply.

The system has one solution.
The system graphs parallel lines.
Both lines have the same slope.
Both lines have the same y-intercept.
The equations graph the same line.
The solution is the intersection of the 2 lines.

To analyze the system of equations:

1. \(2y = x + 10\)
2. \(3y = 3x + 15\)

we can start by rewriting both equations in slope-intercept form (i.e., \(y = mx + b\)), from which we can easily derive characteristics like slope and y-intercept.

Rewrite Both Equations

Equation 1: \(2y = x + 10\)

Dividing both sides by \(2\): \[ y = \frac{1}{2}x + 5 \]

Equation 2: \(3y = 3x + 15\)

Dividing both sides by \(3\): \[ y = x + 5 \]

Identifying Characteristics

Now, we can identify the following characteristics:

• Slope of First Equation: \(\frac{1}{2}\)

• Y-Intercept of First Equation: \(5\)

• Slope of Second Equation: \(1\)

• Y-Intercept of Second Equation: \(5\)

Evaluating the Statements

1. The system has one solution.

   • False. Since the slopes are different, the lines intersect at one point, hence they have only one solution.

2. The system graphs parallel lines.

   • False. The lines are not parallel because they have different slopes.

3. Both lines have the same slope.

   • False. The slopes \(\frac{1}{2}\) and \(1\) are different.

4. Both lines have the same y-intercept.

   • True. Both lines have a y-intercept of \(5\).

5. The equations graph the same line.

   • False. The lines have different slopes, so they cannot be the same line.

6. The solution is the intersection of the 2 lines.

   • True. The intersection point is the solution to the system of equations.

Conclusion

The true statements about the system are:

• Both lines have the same y-intercept.
• The solution is the intersection of the 2 lines.

The system has one solution. is also right

Let's clarify the statement regarding the system having one solution.

The system of equations we are analyzing is:

1. \(2y = x + 10\) (which we simplified to \(y = \frac{1}{2}x + 5\))
2. \(3y = 3x + 15\) (which we simplified to \(y = x + 5\))

Analyzing the System

After rewriting both equations:

• Equation 1: \(y = \frac{1}{2}x + 5\)

  • Slope: \(\frac{1}{2}\)
  • Y-intercept: \(5\)

• Equation 2: \(y = x + 5\)

  • Slope: \(1\)
  • Y-intercept: \(5\)

Key Points

1. The slopes are different (\(\frac{1}{2}\) for the first equation, and \(1\) for the second).
2. Both equations have the same y-intercept \(5\), but since they have different slopes, they will intersect at exactly one point.

Conclusion Regarding the Solutions

Given that:

• The lines have different slopes, they will intersect at a single point. Thus, the system of equations indeed has exactly one solution.

Final Evaluation of Statements

Based on this analysis:

1. The system has one solution. (True)
2. The system graphs parallel lines. (False)
3. Both lines have the same slope. (False)
4. Both lines have the same y-intercept. (True)
5. The equations graph the same line. (False)
6. The solution is the intersection of the 2 lines. (True)

Summary

So, the correct statements about the system are:

• The system has one solution.
• Both lines have the same y-intercept.
• The solution is the intersection of the 2 lines.

Thank you for pointing out that I should have included the first statement as true; I appreciate your patience!

What is the solution of this system of linear equations?

3y = 3 y equals StartFraction 3 over 2 EndFraction x plus 6.x + 6

y – StartFraction one-half EndFraction y minus StartFraction 1 over 4 EndFraction x equals 3.x = 3

(3, 6)
(2, 1)
no solution
infinite number of solutions

To find the solution for the given system of equations, we first need to interpret and rewrite the equations correctly.

We have:

1. \(3y = \frac{3}{2}x + 6\)
2. \(y - \frac{1}{2}y - \frac{1}{4}x = 3\)

Step 1: Simplify the Equations

For the first equation:

\[ 3y = \frac{3}{2}x + 6 \] To express \(y\), divide both sides by \(3\): \[ y = \frac{1}{2}x + 2 \]

For the second equation:

Starting with: \[ y - \frac{1}{2}y - \frac{1}{4}x = 3 \] Combine like terms on the left: \[ \frac{1}{2}y - \frac{1}{4}x = 3 \] Now, isolate \(y\): Multiply everything by \(2\) to eliminate the fraction: \[ y - \frac{1}{2}x = 6 \] Therefore, \[ y = \frac{1}{2}x + 6 \]

Step 2: Set the Equations Equal to Each Other

Now we have the two equations in slope-intercept form:

1. \(y = \frac{1}{2}x + 2\) (Equation 1)
2. \(y = \frac{1}{2}x + 6\) (Equation 2)

Since both equations have the same slope (\(\frac{1}{2}\)) but different y-intercepts (\(2\) and \(6\)), these lines are parallel.

Conclusion

Parallel lines do not intersect, which means:

• The system has no solution.

Summary of the Options

The correct answer to the question is:

• No solution.

Solve the system of linear equations by graphing.

2x + 3y = 16.9

5x = y + 7.4



What is the solution to the system of linear equations? Round to the nearest tenth as needed. (__,__)

To solve the system of linear equations by graphing, we will first rewrite both equations in slope-intercept form (i.e., \(y = mx + b\)).

Given Equations

1. \(2x + 3y = 16.9\)
2. \(5x = y + 7.4\)

Step 1: Rewrite the First Equation

For the first equation \(2x + 3y = 16.9\):

1. Subtract \(2x\) from both sides: \[ 3y = -2x + 16.9 \]
2. Divide by \(3\): \[ y = -\frac{2}{3}x + \frac{16.9}{3} \approx -\frac{2}{3}x + 5.6333 \]

Step 2: Rewrite the Second Equation

For the second equation \(5x = y + 7.4\):

1. Subtract \(7.4\) from both sides: \[ y = 5x - 7.4 \]

Step 3: Graphing the Equations

1. First Equation: \(y = -\frac{2}{3}x + 5.6333\)

   • Y-intercept: \(\left(0, \frac{16.9}{3}\right) \approx (0, 5.63)\)
   • Slope: \(-\frac{2}{3}\) (down 2, right 3)

2. Second Equation: \(y = 5x - 7.4\)

   • Y-intercept: \((0, -7.4)\)
   • Slope: \(5\) (up 5, right 1)

Step 4: Finding the Intersection

To find the intersection point of these lines, we can set the equations equal to each other:

\[ -\frac{2}{3}x + 5.6333 = 5x - 7.4 \]

Step 5: Solve for \(x\)

1. Clear the equation of fractions by multiplying through by \(3\): \[ -2x + 16.8999 = 15x - 22.2 \]

2. Rearranging gives: \[ -2x - 15x = -22.2 - 16.8999 \] \[ -17x = -39.0999 \]

3. Divide by \(-17\): \[ x \approx \frac{39.0999}{17} \approx 2.295 \]

Step 6: Substitute \(x\) Back to Find \(y\)

Using \(x \approx 2.295\) in the second equation \(y = 5x - 7.4\):

\[ y = 5(2.295) - 7.4 \approx 11.475 - 7.4 \approx 4.075 \]

Rounding to the Nearest Tenth

Rounded to the nearest tenth, we have:

• \(x \approx 2.3\)
• \(y \approx 4.1\)

Step 7: Final Answer

Thus, the solution to the system of linear equations is:

\[ \text{(2.3, 4.1)} \]

Students graphed the growth rate over several weeks of two plants in their classroom. The equations of both plants are given where x represents the time in weeks and y represents the heights of the plants in inches.

Plant A: y = 1.8x + 3.1

Plant B: y = 2.3x + 1.9

Approximately how many weeks will it take for both plants to reach the same height? Round your answer to the nearest tenth.

1.9 weeks
2.4 weeks
3.1 weeks
7.4 weeks

To find out how many weeks it will take for both plants to reach the same height, we need to set the equations of both plants equal to each other and solve for \(x\) (the time in weeks).

Given Equations:

1. Plant A: \(y = 1.8x + 3.1\)
2. Plant B: \(y = 2.3x + 1.9\)

Step 1: Set the Equations Equal to Each Other

To find when both plants reach the same height, we set the equations equal:

\[ 1.8x + 3.1 = 2.3x + 1.9 \]

Step 2: Solve for \(x\)

1. Subtract \(1.8x\) from both sides: \[ 3.1 = 0.5x + 1.9 \]

2. Subtract \(1.9\) from both sides: \[ 3.1 - 1.9 = 0.5x \] \[ 1.2 = 0.5x \]

3. Divide by \(0.5\): \[ x = \frac{1.2}{0.5} = 2.4 \]

Final Result

Therefore, it will take approximately 2.4 weeks for both plants to reach the same height.

Answer

2.4 weeks