Graphing Linear Equations and Solving Systems of Equations

Graphing Linear Equations

This section explains how to graph linear equations and determine the solutions for a system of equations.

Equations Given

The two equations that need to be graphed are:

1. $y = x + 4$

2. $y = -x + 9$

Equation 1: $y = x + 4$

Characteristics:

• This is a linear equation in slope-intercept form, where the slope (m) is 1 and the y-intercept (b) is 4.

• The graph will intersect the y-axis at (0, 4).

• To find another point, when $x = 1$, then $y = 1 + 4 = 5$; therefore, the point (1, 5) is also on the line.

• Additional points can be found by substituting other values for $x$:

  • When $x = -1$, $y = -1 + 4 = 3$ → Point (-1, 3).

  • When $x = 2$, $y = 2 + 4 = 6$ → Point (2, 6).

Equation 2: $y = -x + 9$

Characteristics:

• This also is a linear equation in slope-intercept form, where the slope (m) is -1 and the y-intercept (b) is 9.

• The graph will intersect the y-axis at (0, 9).

• To find another point, when $x = 1$, then $y = -1 + 9 = 8$; therefore, the point (1, 8) is also on the line.

• Additional points can be found by substituting other values for $x$:

  • When $x = 0$, $y = 9$ → Point (0, 9).

  • When $x = 2$, $y = -2 + 9 = 7$ → Point (2, 7).

Graphing the Equations

• Graphing Process: Begin by plotting the points found for each equation on a coordinate system (x-y plane). Connect these points to form straight lines for each equation. The first line represents the equation $y = x + 4$ and the second line represents $y = -x + 9$.

Determining the Solution of the System

The solution to a system of equations occurs at the point where the two lines intersect.

Finding the Intersection:

To find the coordinates of the solution (intersection), we can set the two equations equal:

• Setting the two equations:
  $x + 4 = -x + 9$

• Solving for $x$:

  1. Add $x$ to both sides:
     $x + x + 4 = 9$

  2. Combine like terms:
     $2x + 4 = 9$

  3. Subtract 4 from both sides:
     $2x = 5$

  4. Divide by 2:
     $x = rac{5}{2}$

• Now substitute $x$ back into either of the original equations to find $y$:
  Using equation 1:
  $y = rac{5}{2} + 4 = rac{5}{2} + rac{8}{2} = rac{13}{2}$

Solution of the System

• The point of intersection (solution of the system of equations) is therefore:
  $ext{Solution: } \left( \frac{5}{2}, \frac{13}{2} \right)$

• Note that this corresponds to the ordered pair ((2.5, 6.5)).

Conclusion

• The graphical representation of these two equations demonstrates the method for solving systems of equations. The ordered pair found from the intersection represents the single solution to the given system of equations, as indicated by the point where the two lines meet on the graph.

• Hence, the solution to the system of equations is:
  $\left( \frac{5}{2}, \frac{13}{2} \right)$ or more simply written as $\left( 2.5, 6.5 \right)$.