Find the solution of the given equation in graphical method

8x + 5y = 2
-2x + y = 4

Given:1. [tex]8x + 5y = 2[/tex]2. [tex]-2x + y = 4[/tex]Step 1: Rearrange each equation for [tex]$$y$$[/tex] (slope-intercept form).For equation (1):     [tex]8x + 5y = 2 \implies 5y = 2 - 8x \implies y = \frac{2 - 8x}{5} = \frac{2}{5} - \frac{8}{5}x[/tex]For equation (2):     [tex]-2x + y = 4 \implies y = 4 + 2x[/tex]Step 2: Plot both equations on the coordinate plane.For [tex]y = \frac{2}{5} - \frac{8}{5}x[/tex]:When [tex]x = 0, y = \frac{2}{5} = 0.4[/tex]When [tex]x = 1, y = \frac{2}{5} - \frac{8}{5} = -\frac{6}{5} = -1.2[/tex]For [tex]$$y = 4 + 2x$$[/tex]:When [tex]x = 0, y = 4[/tex]When [tex]x = 1, y = 4 + 2 = 6[/tex]Step 3: Draw both lines on the graph using these points.Step 4: Identify the point of intersection of the two lines.The point where the two lines intersect is the solution to the system of equations.Step 5: To find the exact solution algebraically, set the two expressions for [tex]$$y$$[/tex] equal:     [tex]\frac{2}{5} - \frac{8}{5}x = 4 + 2x[/tex]Multiply both sides by 5 to clear fractions:     [tex]2 - 8x = 20 + 10x[/tex]Bring variables to one side:     [tex]2 - 8x - 20 - 10x = 0 \implies -18x - 18 = 0[/tex]Simplify:    [tex]-18x = 18 \implies x = -1[/tex]Solve for [tex]$$y$$[/tex]:    [tex]y = 4 + 2(-1) = 4 - 2 = 2[/tex]So, the solution to the system is [tex]$$(-1, 2)$$[/tex]. This is the point where the lines intersect on the graph.Therefore, the solution of the system [tex]$$8x + 5y = 2$$[/tex] and [tex]$$-2x + y = 4$$[/tex] is [tex]$$x = -1$$[/tex], [tex]$$y = 2$$[/tex].

Answered by Sefton | 2025-08-25