hint graph/plot the two points { 2 points} which equation represents the line that passes through the points…

Explanation:

Step1: Calculate the slope

The slope ( m ) between two points ((x_1, y_1)) and ((x_2, y_2)) is given by ( m=\frac{y_2 - y_1}{x_2 - x_1} ). For the points ((-1, 8)) and ((4, -2)), we have ( x_1=-1, y_1 = 8, x_2 = 4, y_2=-2 ). So, ( m=\frac{-2 - 8}{4-(-1)}=\frac{-10}{5}=-2 ).

Step2: Use point - slope form to find the equation

The point - slope form of a line is ( y - y_1=m(x - x_1) ). We can use the point ((-1, 8)) and ( m=-2 ). Substituting these values into the formula: ( y - 8=-2(x+1) ). Expand the right - hand side: ( y - 8=-2x-2 ). Add 8 to both sides: ( y=-2x - 2 + 8=-2x + 6 )? Wait, no, wait. Wait, ( y-8=-2(x + 1)) expands to ( y-8=-2x-2 ), then ( y=-2x-2 + 8=-2x + 6 )? But let's check with the other point. Let's take ( x = 4 ) in ( y=-2x + 6 ), then ( y=-8 + 6=-2 ), which matches the point ((4,-2)). Wait, but let's check the first point: when ( x=-1 ), ( y=-2\times(-1)+6=2 + 6 = 8 ), which matches ((-1,8)). Wait, but let's check the options again. Wait, option (1) is ( y=-2x + 6 ), option (2) is ( y=-2x + 10 ). Wait, maybe I made a mistake. Wait, let's re - calculate the slope. Wait, ( (y_2 - y_1)=-2-8=-10 ), ( (x_2 - x_1)=4-(-1)=5 ), so slope ( m=-2 ). Then using point - slope with ((-1,8)): ( y-8=-2(x + 1)), ( y=-2x-2 + 8=-2x + 6 ). Let's check the point ((4,-2)): ( y=-2\times4+6=-8 + 6=-2 ), which is correct. Wait, but let's check option (2): when ( x=-1 ), ( y=-2\times(-1)+10=2 + 10 = 12\neq8 ). So the correct equation should be ( y=-2x + 6 )? But wait, no, wait, let's check the calculation again. Wait, ( y-8=-2(x + 1)): ( y=-2x-2 + 8=-2x + 6 ). Yes, that's correct. Wait, but let's check the options. Option (1) is ( y=-2x + 6 ), which satisfies both points. Wait, but let's check the other way. Let's substitute ( x=-1 ) into option (2): ( y=-2\times(-1)+10=2 + 10 = 12\neq8 ). Substitute ( x = 4 ) into option (2): ( y=-8 + 10 = 2\neq-2 ). Substitute ( x=-1 ) into option (1): ( y=-2\times(-1)+6=2 + 6 = 8 ), which is correct. Substitute ( x = 4 ) into option (1): ( y=-8 + 6=-2 ), which is correct. Wait, but earlier I thought maybe I made a mistake, but this seems correct. Wait, but let's check the slope again. The two points are ((-1,8)) and ((4,-2)). The change in ( y ) is (-2-8=-10 ), change in ( x ) is (4-(-1)=5 ), slope is (-2). Then using ( y=mx + b ), plug in ( x=-1,y = 8,m=-2 ): ( 8=-2\times(-1)+b\Rightarrow8 = 2 + b\Rightarrow b = 6 ). So the equation is ( y=-2x + 6 ), which is option (1). Wait, but let's check the options again. The options are (1) ( y=-2x + 6 ), (2) ( y=-2x + 10 ), (3) ( y=-0.5x + 7.5 ), (4) ( y=-0.5x + 8.5 ). So the correct equation is ( y=-2x + 6 ), which is option (1).

Wait, no, wait a second. Wait, when ( x=-1 ), ( y=-2\times(-1)+6=2 + 6 = 8 ) (correct). When ( x = 4 ), ( y=-2\times4+6=-8 + 6=-2 ) (correct). So the equation is ( y=-2x + 6 ), which is option (1).

Answer:

(1) ( y=-2x + 6 )