To estimate the solution to the system of equations given by:
- \( y = -x + 6 \)
- \( y = 3z - 2 \)
we can follow these steps:
Step 1: Understanding the Equations
-
Equation 1: This is a linear equation with a y-intercept of 6. It indicates that for each unit increase in \( x \), \( y \) decreases by 1.
-
Equation 2: This states that \( y \) varies with \( z \). Specifically, \( y \) increases by 3 for each unit increase in \( z \) and has a y-intercept of -2.
Step 2: Finding Intersections
To find the solution to the system of equations, we need to find a common solution for \( y \) in terms of both \( x \) and \( z \).
Setting the two expressions for \( y \) equal to each other:
\[ -x + 6 = 3z - 2 \]
Step 3: Rearranging the Equation
Rearranging this equation gives:
\[ -x - 3z = -8 \] \[ x + 3z = 8 \]
Step 4: Analyzing the Relationship
This equation represents a plane in \( xz \)-coordinate system. To describe solutions, we can express \( z \) in terms of \( x \):
\[ 3z = 8 - x \] \[ z = \frac{8 - x}{3} \]
Step 5: Graphing to Estimate Intersection
If you were to graph these equations:
- The first equation, \( y = -x + 6 \), is a straight line declining from 6 on the y-axis.
- The second equation, \( y = 3z - 2 \), represents another line, but it's expressed in terms of \( z \) rather than \( x \).
Conclusion
To find specific numeric values, values of \( x \) or \( z \) need to be specified or the graph needs to be analyzed directly. However, from the derived equations, you can extract various pairs of \( (x, y, z) \) that satisfy both equations.
If you plot both of these equations on a coordinate graph, the solution to the system would be at the point where both lines intersect.
If you have access to a graph, see where the two lines intersect to give you a visual estimation of the solution. If you provide specific graph coordinates or additional information, we can further refine the estimation.