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

For a point to be a solution to a linear equation, the coordinates of that point must satisfy the equation when substituted into it. In other words, if you have a point represented as \((x, y)\), it is a solution to the equation \(y = mx + b\) if, when you plug in the \(x\) value into the equation, you get the corresponding \(y\) value.

In your example, you claim that the point \((2, 5)\) is a solution to the equation \(y = 2x + 3\). To check this claim, follow these steps:

1. Identify the coordinates of the point: Here, \(x = 2\) and \(y = 5\).

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

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

4. Compare the calculated \(y\) value to the \(y\) coordinate of the point: You have found that when \(x = 2\), \(y\) should be 7, but the point states that \(y\) is 5.

Since the calculated \(y\) value (7) does not equal the \(y\) coordinate of the point (5), \((2, 5)\) is not a solution to the equation \(y = 2x + 3\).

In summary, to determine if a point is a solution, substitute the \(x\) value from the point into the equation and check if the output matches the \(y\) value of the point.

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 find an ordered pair that is a solution to both equations \(y = x + 1\) and \(y = -x + 5\), we can set the equations equal to each other and solve for \(x\):

1. Set the two equations equal to each other: \[ x + 1 = -x + 5 \]

2. Solve for \(x\): \[ x + x = 5 - 1 \] \[ 2x = 4 \] \[ x = 2 \]

3. Substitute \(x = 2\) back into one of the equations to find \(y\). We'll use \(y = x + 1\): \[ y = 2 + 1 = 3 \]

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

Verification:

• For \(y = x + 1\): \[ y = 2 + 1 = 3 \] (True)

• For \(y = -x + 5\): \[ y = -2 + 5 = 3 \] (True)

Thus, \((2, 3)\) satisfies both equations. You can also graph both equations using a tool like Desmos to visually confirm that they intersect at the point \((2, 3)\).