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Method

Solving a System of Linear Equations Graphically

Solving a system of linear equations graphically means graphing the lines represented by the equations of the system and identifying the point of intersection. Consider an example system of equations.

{2 y = - 2 x + 8 x = y - 1

To solve the system of equations, three steps must be followed.

Write the Equations in Slope-Intercept Form

Start by writing the equations in slope-intercept form by isolating the

y -

variables. For the first linear equation, divide both sides by

2 .

For the second equation, add

1

to both sides.

{2 y = - 2 x + 8 x = y - 1 (I) (II)

(I):

Solve for

y

DivEqn

$(I):$ $LHS / 2 = RHS / 2$

{y = \frac{- 2 x + 8}{2} x = y - 1

WriteSumFrac

$(I):$ Write as a sum of fractions

{y = \frac{- 2 x}{2} + \frac{8}{2} x = y - 1

MovePartNumRight

$(I):$ $\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

{y = \frac{- 2}{2} x + \frac{8}{2} x = y - 1

MoveNegNumToFrac

$(I):$ Put minus sign in front of fraction

{y = - \frac{2}{2} x + \frac{8}{2} x = y - 1

CalcQuot

$(I):$ Calculate quotient

{y = - 1 x + 4 x = y - 1

IdPropMult

$(I):$ Identity Property of Multiplication

{y = - x + 4 x = y - 1

(II):

Solve for

y

AddEqn

$(II):$ $LHS + 1 = RHS + 1$

{y = - x + 4 x + 1 = y

RearrangeEqn

$(II):$ Rearrange equation

{y = - x + 4 y = x + 1

Graph the Lines

Now that the equations are both written in slope-intercept form, they can be graphed on the same coordinate plane.

Graphs of two lines using the slopes and y-intercepts

Identify the Point of Intersection

The point where the lines intersect is the solution to the system.

The lines appear to intersect at $(1.5, 2.5) .$ Therefore, this is the solution to the system — the value of $x$ is $1.5$ and the value of $y$ is $2.5 .$

Sometimes the point of intersection of the lines is not a lattice point. In this case, the solution found by solving the system of equations graphically is approximate.

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