Solve by graphing.If there isn't a unique solution to the system,state the reason. A) x-y=3 x+y=5   B) 4x+y=5  3x-2y=12

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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A) You need to solve for x and y the system of equations, using graphing, hence, you need to sktech the graphs of the lines represented by the given equations and check if they intersect, such that:

`{(x-y=3),(x+y=5):}`

You need to convert equation in slope intercept form, hence, you need to isolate y to the left side, such that:

`{(y = x - 3),(y = - x + 5):}`

You may find the intercepting point such that:

`x - 3 = -x + 5 => 2x = 5+3 => 2x = 8 => x = 4`

`y = 4 - 3 => y = 1`

You need to select a point that lies on the line y = x - 3 such that:

x = 1 => y = 1 - 3 = -2

You need to trace a line that passes through the points (1,-2) and (4,1).

You need to select a point that lies on the line y = -x + 5 such that:

`x = 1 => y = 4 `

You need to trace a line that passes through the points (1,4) and (4,1).

These two lines share a common point whose coordinates are `x = 4, y = 1` . The point (4,1) represents the solution to the given system of equations.

You need to sketch the graphs of the lines, such that:

The lines are intercepting at `x = 4, y = 1` , hence, the solution to the system is `x = 4, y = 1.`

B) You need to perform the same steps as you did at A), such that:

`{(4x+y=5),(3x-2y=12):} => {(y = -4x + 5),(2y = 3/2x - 6):} `

The lines intersect each other at `x = 2, y = -3` , hence, the solution to the system is `x = 2, y = -3.`

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ishpiro | College Teacher | (Level 1) Educator

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   A)

For problem A), use the intercepts to graph both equations. You can find the intercepts using the cover-up method:

x-y = 3

If you cover up x (set x = 0) that gives you -y=3, or y = -3

This is y-intercept (0, -3)

If you cover up y (set y = 0) that gives you x= 3

This is x-intercept (3,0)

Plot these points on the coordinate plane and connect them to get the straigh line - the graph of the first equation.

Similarly, the intercepts for the second equation will be (0, 5) and (5, 0). Plot them and connect them with the straight line - the graph of the second equation.

As you can see on the graph above, the line intersects at (4, 1).

This is the solution to the first system: x = 4, y = 1

 

B)

For problem B), use the slope-intercept method to graph the first equation:

Rewrite 4x + y = 5 as y = -4x + 5 by subtracting 4x from both sides. Now it is in slope-intercept form and you can identify slope as - 4 and y-intercept as (0, 5). Plot (0, 5) first. Then, since slope is -4, find the second point by moving 1 unit to the right and 4 units DOWN. Connect this point to the (0,5) with the straight line - the graph of the first equation.

The second equation is convinient to graph using intercepts:

3x - 2y = 12

If x = 0 then y = -6

The y-intercept is (0, -6)

If y = 0 then x = 4

The x-intercept is (4, 0)

Plot these points and connect them with the straight line - the graph of the second equation.

As you can see on the graph above, the two lines intersect at (2, -3). So the solution to B) is x = 2, y = -3.

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