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Solve the system of equations by graphing.

How many solutions?

y =-2x + 3
-2x = y +1

11 answers

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

  1. Equation 1: \[ y = -2x + 3 \] This is already in slope-intercept form.

  2. Equation 2: Starting with the equation: \[ -2x = y + 1 \] We can rearrange it to solve for \(y\): \[ y = -2x - 1 \]

Now we have the two equations:

  1. \(y = -2x + 3\)
  2. \(y = -2x - 1\)

Next, we will graph both equations.

Graphing the Equations:

  1. For the first equation \(y = -2x + 3\):

    • When \(x = 0\), \(y = 3\) (point: (0, 3))
    • When \(x = 1\), \(y = 1\) (point: (1, 1))
    • When \(x = 2\), \(y = -1\) (point: (2, -1))

    Plot these points and draw a line through them.

  2. For the second equation \(y = -2x - 1\):

    • When \(x = 0\), \(y = -1\) (point: (0, -1))
    • When \(x = 1\), \(y = -3\) (point: (1, -3))
    • When \(x = 2\), \(y = -5\) (point: (2, -5))

    Plot these points and draw a line through them.

Analyzing the Graphs:

Upon graphing both equations, we can observe:

  • The first line \(y = -2x + 3\) has a slope of \(-2\) and intercepts the y-axis at \(3\).
  • The second line \(y = -2x - 1\) also has a slope of \(-2\) but intercepts the y-axis at \(-1\).

Conclusion:

Both lines have the same slope but different y-intercepts, which means they are parallel. Since parallel lines never intersect, the system of equations has no solutions.

Final Answer:

Number of solutions: 0 (no solutions).

Solve the system of equations by graphing.

How many solutions are you expecting?
Why?
X = 4
Y = 2x - 3

What do you notice about the line
x = 4 ?

What equations would make a horizontal line?

To solve the system of equations by graphing, we first identify and analyze the equations given.

  1. Equation 1: \[ x = 4 \] This represents a vertical line where the x-coordinate is always 4. It passes through all points where \(x = 4\).

  2. Equation 2: \[ y = 2x - 3 \] This is already in slope-intercept form, where the slope \(m = 2\) and the y-intercept \(b = -3\).

Graphing the Equations:

  • For \(x = 4\):

    • This is a vertical line that runs through all points where \(x = 4\) (the line itself is vertical).

  • For \(y = 2x - 3\):

    • When \(x = 0\), \(y = -3\) (point: (0, -3)).
    • When \(x = 1\), \(y = -1\) (point: (1, -1)).
    • When \(x = 2\), \(y = 1\) (point: (2, 1)).
    • When \(x = 3\), \(y = 3\) (point: (3, 3)).
    • When \(x = 4\), \(y = 5\) (point: (4, 5)).

    Plot these points and draw the line through them.

Analyzing the Graphs:

  • The vertical line \(x = 4\) passes through (4, y) for any value of y.
  • The linear equation \(y = 2x - 3\) creates an upward sloping line that intersects the vertical line.

To find the intersection:

  • Substitute \(x = 4\) into the second equation: \[ y = 2(4) - 3 = 8 - 3 = 5. \]
  • So the intersection point is \((4, 5)\).

Expectation for Solutions:

How many solutions are you expecting? Why?

  • I am expecting 1 solution, as the vertical line \(x = 4\) intersects the line \(y = 2x - 3\) at exactly one point, which is (4, 5).

Noticing about the line \(x = 4\):

The line \(x = 4\) is a vertical line. It indicates that for any value of \(y\), the corresponding \(x\) value is always 4. Vertical lines are defined by a constant \(x\) value.

What equations would make a horizontal line?

A horizontal line is defined by a constant \(y\) value, meaning \(y\) does not change regardless of the \(x\) value. The general form of a horizontal line is: \[ y = b, \] where \(b\) is a constant. For example, \(y = 3\) would represent a horizontal line where \(y\) is always 3, regardless of the value of \(x\).

Summary:

Number of solutions: 1 (the intersection point \((4, 5)\)).
The line \(x = 4\) is vertical, and equations of the form \(y = b\) represent horizontal lines.

12(x – 3) + 18 =

To simplify the expression \(12(x - 3) + 18\), we can follow these steps:

  1. Distribute \(12\) to the terms inside the parentheses: \[ 12(x - 3) = 12x - 36 \]

  2. Now, substitute this back into the expression: \[ 12(x - 3) + 18 = 12x - 36 + 18 \]

  3. Combine the constant terms \(-36\) and \(18\): \[ -36 + 18 = -18 \]

  4. Thus, the simplified expression is: \[ 12x - 18 \]

So, the final result of the expression \(12(x - 3) + 18\) is:

\[ \boxed{12x - 18} \]

how many solutions does 6(2x -3 )

The expression \(6(2x - 3)\) is not an equation, so it does not have solutions in the way that equations do. Instead, it can be simplified or evaluated.

However, if you intended to ask how many solutions the equation \(6(2x - 3) = 0\) has, we can solve that:

  1. Set the expression equal to zero: \[ 6(2x - 3) = 0 \]

  2. Divide both sides by 6 (since 6 is not zero, we can do this): \[ 2x - 3 = 0 \]

  3. Add 3 to both sides: \[ 2x = 3 \]

  4. Divide both sides by 2: \[ x = \frac{3}{2} \]

Conclusion:

The equation has 1 solution, which is: \[ x = \frac{3}{2} \]

If you meant something else regarding the expression, please clarify!

how many solutions
4(3x - 3)

The expression \(4(3x - 3)\) is an algebraic expression and does not represent an equation. Therefore, it does not have solutions in the typical sense. It can be simplified, but there are no "solutions" to find unless it is set equal to something (like zero).

If you're asking about how many solutions the equation \(4(3x - 3) = 0\) has, we can solve that as follows:

  1. Set the expression equal to zero: \[ 4(3x - 3) = 0 \]

  2. Divide both sides by 4: \[ 3x - 3 = 0 \]

  3. Add 3 to both sides: \[ 3x = 3 \]

  4. Divide both sides by 3: \[ x = 1 \]

Conclusion:

The equation \(4(3x - 3) = 0\) has 1 solution, which is: \[ x = 1 \]

If you meant something else regarding the expression, please let me know!

how many solutions
4(2x - 3)

The expression \(4(2x - 3)\) by itself does not represent an equation, so it does not have solutions in the sense of finding values for \(x\). It is an algebraic expression that can be simplified, but it does not equal anything unless specified.

However, if you are asking about how many solutions the equation \(4(2x - 3) = 0\) has, we can solve that:

  1. Set the equation equal to zero: \[ 4(2x - 3) = 0 \]

  2. Divide both sides by 4 (valid since 4 is not zero): \[ 2x - 3 = 0 \]

  3. Add 3 to both sides: \[ 2x = 3 \]

  4. Divide both sides by 2: \[ x = \frac{3}{2} \]

Conclusion:

The equation \(4(2x - 3) = 0\) has 1 solution, which is: \[ x = \frac{3}{2} \]

If you have another type of question regarding the expression, please clarify!