Q:

1. Write the standard form of the line that passes through the given points. Include your work in your final answer. (3, 1) and (-2, 3)2. (4, 7) and (0, 7)3. (2, 3) and (2, 5)4.Write the slope-intercept form of the line with a slope of 2 and a y -intercept of -4. Include your work in your final answer. 5. Write the standard form of the line that is parallel to 2 x + 3 y = 4 and passes through the point (1, -4). Include your work in your final answer.6. Write the standard form of the line that contains a slope of 2/3 and passes through the point (1, 1). Include your work in your final answer.

Accepted Solution

A:
Answer:Part 1) [tex]2x+5y=11[/tex] Part 2) [tex]y=7[/tex]Part 3) [tex]x=2[/tex]Part 4) [tex]y=2x-4[/tex]Part 5) [tex]2x+3y=-10[/tex]Part 6) [tex]2x-3y=-1[/tex]Step-by-step explanation:Part 1) Write the standard form of the line that passes through the given points(3, 1) and (-2, 3)we know thatThe equation of the line in standard form is equal to Ax+By=CwhereA is a positive integerB and C are integersstep 1Find the slope m[tex]m=(3-1)/(-2-3)[/tex][tex]m=-2/5[/tex]step 2Find the equation in point slope form[tex]y-y1=m(x-x1)[/tex]we have[tex]m=-2/5[/tex] and point [tex](3,1)[/tex]substitute[tex]y-1=-(2/5)(x-3)[/tex][tex]y=-(2/5)x+(6/5)+1[/tex][tex]y=-(2/5)x+(11/5)[/tex]Convert to standard formMultiply by 5 both sides[tex]5y=-2x+11[/tex][tex]2x+5y=11[/tex] -----> equation in standard formPart 2) Write the standard form of the line that passes through the given points(4, 7) and (0, 7)we know thatThe equation of the line in standard form is equal to Ax+By=CwhereA is a positive integerB and C are integersstep 1Find the slope m[tex]m=(7-7)/(0-4)=0[/tex]This is a horizontal line (parallel to the x-axis)The equation of the line is [tex]y=7[/tex]Part 3) Write the standard form of the line that passes through the given points(2, 3) and (2, 5)we know thatThe equation of the line in standard form is equal to Ax+By=CwhereA is a positive integerB and C are integersstep 1Find the slope m[tex]m=(5-3)/(2-2)[/tex] [tex]m=2/0[/tex] ----> the slope is undefined This is a vertical line (parallel to the y-axis)The equation of the line is [tex]x=2[/tex]Part 4) Write the slope-intercept form of the line with a slope of 2 and a y -intercept of -4.we know thatThe equation of the line into slope-intercept form is equal to[tex]y=mx+b[/tex]wherem is the slope and b is the y-interceptwe have[tex]m=2[/tex][tex]b=-4[/tex]substitute[tex]y=2x-4[/tex]Part 5) Write the standard form of the line that is parallel to 2 x + 3 y = 4 and passes through the point (1, -4). we know thatIf two lines are parallel, then their slopes are the samewe have[tex]2x+3y=4[/tex]isolate the variable y[tex]3y=4-2x[/tex][tex]y=(4/3)-(2/3)x[/tex]The slope of the given line is [tex]m=-2/3[/tex]soFind the equation of the line with slope m=-2/3 and passes through the point (1,-4)[tex]y-y1=m(x-x1)[/tex]substitute[tex]y+4=-(2/3)(x-1)[/tex][tex]y=-(2/3)x+(2/3)-4[/tex][tex]y=-(2/3)x-(10/3)[/tex]Convert to standard formMultiply by 3 both sides[tex]3y=-2x-10[/tex][tex]2x+3y=-10[/tex]Part 6) Write the standard form of the line that contains a slope of 2/3 and passes through the point (1, 1)Find the equation in point slope form[tex]y-y1=m(x-x1)[/tex]we have[tex]m=2/3[/tex] and point [tex](1,1)[/tex]substitute[tex]y-1=(2/3)(x-1)[/tex][tex]y=(2/3)x-(2/3)+1[/tex][tex]y=(2/3)x+(1/3)[/tex]Multiply by 3 both sides[tex]3y=2x+1[/tex][tex]2x-3y=-1[/tex]