To create a consistent and independent system of equations, we need to find an equation that intersects with the given equation 3x + 4y = 8 at a single point. This means the two equations should have different slopes.
One way to form such an equation is to rewrite the first in slope-intercept form (y = mx + b). The original equation can be rearranged:
4y = -3x + 8
y = -\frac{3}{4}x + 2The slope of this line is -3/4. To ensure the system is consistent and independent, we should choose an equation with a different slope. For example, we could use:
2x + y = 5Rearranging it into slope-intercept form gives:
y = -2x + 5The slope is -2, which is different from -3/4. Therefore, this equation, 2x + y = 5, together with 3x + 4y = 8, forms a consistent and independent system, as they intersect at exactly one point.