Put a point on the line.

To find a point on the line with an x-intercept of -4.5 and a y-intercept of 9, we can use the intercepts to determine the linear equation of the line.

1. The x-intercept (-4.5, 0) means that when \( y = 0 \), \( x = -4.5 \).
2. The y-intercept (0, 9) means that when \( x = 0 \), \( y = 9 \).

Now we can use these two points to find the equation of the line in slope-intercept form \( y = mx + b \).

The slope \( m \) can be calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 0}{0 - (-4.5)} = \frac{9}{4.5} = 2 \]

Now we have the slope, and we can use the y-intercept to write the equation: \[ y = 2x + 9 \]

We can now choose an x-value to find a corresponding y-value that lies on the line. For example, let’s use \( x = -4 \): \[ y = 2(-4) + 9 = -8 + 9 = 1 \]

Thus, the ordered pair that is a point on the line is: \[ \text{(-4, 1)} \]

You can also choose other x-values, like -5 or 0, to find different points on the line. For instance, using \( x = 0 \) gives us \( (0, 9) \), which is the y-intercept. Another point can be \( (-4.5, 0) \), which is the x-intercept.

So one valid ordered pair is: \[ (-4, 1) \]

Find a solution to this equation.

y = x + 2
Type an (x y)
ordered pair that is a solution to this equation.

To find a solution to the equation \( y = x + 2 \), we can choose any value for \( x \) and then calculate \( y \) using that equation.

Let's choose \( x = 1 \): \[ y = 1 + 2 = 3 \]

Therefore, the ordered pair that is a solution to the equation is: \[ (1, 3) \]

You can pick any other value for \( x \) as well, such as \( x = 0 \): \[ y = 0 + 2 = 2 \] This gives another solution: \[ (0, 2) \]

So one possible solution is: \[ (1, 3) \]

Your Point
The graph shows a coordinate plane that goes from negative 8 to 8 in each direction. The graph also shows the line "y" equals "x" plus 2.The point you entered on Screen 2, ( 1 , 3 ), is on the line.
Here is your ordered pair plotted as a point, together with the linear equation
.

(If you can't see your point, maybe you didn't type it correctly, or maybe it's outside the graph window.)

Is your ordered pair correct? How do you know?

Yes, the ordered pair \((1, 3)\) is indeed correct for the equation \(y = x + 2\).

To verify that this point lies on the line described by the equation, we can substitute the \(x\) value of the ordered pair into the equation and check if it yields the corresponding \(y\) value.

1. The equation is \(y = x + 2\).
2. Substitute \(x = 1\): \[ y = 1 + 2 = 3 \]

Since substituting \(x = 1\) gives us \(y = 3\), which matches the \(y\) value in the ordered pair \((1, 3)\), we can confirm that the point lies on the line represented by the equation.

This process of substitution is how we ensure that an ordered pair is a solution to a given equation. Therefore, the ordered pair \((1, 3)\) is correct.

What does "solution" mean?
What does it mean for a point to be a solution to a linear equation?

For example, if I say, (2,5)"
is a solution to the equation y = 2x + 3
," how could you check my claim?

In the context of equations, a solution refers to an ordered pair \((x, y)\) that satisfies the equation when substituted into it. This means that when you plug in the \(x\) value of the ordered pair into the equation, the resulting \(y\) value should match the \(y\) value given in the ordered pair.

For a point to be a solution to a linear equation, it must make the equation true when you replace the variables \(x\) and \(y\) with the values from the ordered pair. If both sides of the equation are equal after substitution, then that point is indeed a solution.

Checking the Claim for the Example \((2, 5)\):

Given the equation \(y = 2x + 3\), we want to check if the point \((2, 5)\) is a solution.

1. Identify the \(x\) and \(y\) values in the ordered pair:

   • \(x = 2\)
   • \(y = 5\)

2. Substitute \(x = 2\) into the equation \(y = 2x + 3\): \[ y = 2(2) + 3 \]

3. Calculate the right-hand side: \[ y = 4 + 3 = 7 \]

4. Compare the calculated \(y\) value (7) with the \(y\) value from the ordered pair (5):

   • Since \(7 \neq 5\), the ordered pair \((2, 5)\) is not a solution to the equation \(y = 2x + 3\).

Summary:

To summarize, a point \((x, y)\) is a solution to a linear equation if substituting \(x\) into the equation results in \(y\). In this case, since the calculated value did not match the given \(y\) value, \((2, 5)\) is not a solution to the equation.

Look at these two lines.
The graph shows a coordinate plane that goes from negative 10 to 6 in the horizontal direction and negative 8 to 8 in the vertical direction. The graph also shows two lines. Each line passes through ( negative 4 , 1 ). One line has positive slope. The other line has negative slope.
Is there an ordered pair that is a solution to the linear equations describing BOTH of these lines?

To determine if there is an ordered pair that is a solution to the linear equations of both lines passing through the point \((-4, 1)\) with one line having a positive slope and the other line having a negative slope, we can analyze the situation.

Key Points:

1. Point of Intersection: Both lines intersect at the point \((-4, 1)\). Since this point lies on both lines, it is a solution to the equations of both lines.

2. Characteristics of the Lines:

   • The line with positive slope will increase as you move from left to right.
   • The line with negative slope will decrease as you move from left to right.

Given that both lines intersect at \((-4, 1)\), this ordered pair is definitely a solution to both linear equations.

Conclusion:

Thus, the ordered pair \((-4, 1)\) is a solution to the linear equations describing both lines. If you're looking for other points, those would only exist if the lines intersect again, but since each line has a different slope and they are not parallel, the only point that satisfies both equations is \((-4, 1)\).

Look at these two equations.
Is there an ordered pair that is a solution to BOTH of these linear equations?

*You can type them into the Desmos calculator to see the graph.

y=x+1

y=−x+5

To determine if there is an ordered pair that is a solution to both equations

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

we can set the two equations equal to each other, since they both equal \(y\).

Step 1: Set Equations Equal to Each Other

\[ x + 1 = -x + 5 \]

Step 2: Solve for \(x\)

1. Add \(x\) to both sides: \[ 2x + 1 = 5 \]
2. Subtract 1 from both sides: \[ 2x = 4 \]
3. Divide by 2: \[ x = 2 \]

Step 3: Substitute \(x\) Back to Find \(y\)

Now that we have found \(x\), we can substitute it back into either equation to find \(y\). We'll use the first equation:

\[ y = x + 1 \] \[ y = 2 + 1 = 3 \]

Conclusion

Thus, the ordered pair that is a solution to both equations is \((2, 3)\).

Verification

To verify, we can check this point in both equations:

1. For \(y = x + 1\): \[ y = 2 + 1 = 3 \quad \text{(True)} \]

2. For \(y = -x + 5\): \[ y = -2 + 5 = 3 \quad \text{(True)} \]

Thus, \((2, 3)\) is indeed a solution to both equations.