Unlocking Graphing Systems of Equations: Solving Systems by Graphing Techniques

High School Sport	Number of Students Participating in $2015$ (Thousands)	Average Rate of Increase (Thousands per Year)
Basketball	$700$	$20$
Soccer	$400$	$40$

High School Sport

Number of Students Participating in

2015

(Thousands)

Average Rate of Increase
(Thousands per Year)

Basketball

700

20

Soccer

400

40

a The system will be solved graphically. To do so, both linear equations must to be expressed in slope-intercept form. In the first equation, the variable

y

is already isolated. Therefore, only the second equation needs to be rewritten.

{y = - 2 x + 6 4 x - 2 y = - 20 (I) (II)

▼

(II):

Write in slope-intercept form

SubEqn

$(II):$ $LHS - 4 x = RHS - 4 x$

{y = - 2 x + 6 - 2 y = - 4 x - 20

DivEqn

$(II):$ $LHS / (- 2) = RHS / (- 2)$

{y = - 2 x + 6 y = \frac{- 4 x - 2 0}{- 2}

WriteDiffFrac

$(II):$ Write as a difference of fractions

{y = - 2 x + 6 y = \frac{- 4 x}{- 2} - \frac{2 0}{- 2}

DivNegNeg

$(II):$ $\frac{- a}{- b} = \frac{a}{b}$

{y = - 2 x + 6 y = \frac{4 x}{2} - \frac{2 0}{- 2}

MovePartNumRight

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

{y = - 2 x + 6 y = \frac{4}{2} x - \frac{2 0}{- 2}

MoveNegDenomToFrac

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

{y = - 2 x + 6 y = \frac{4}{2} x - (- \frac{2 0}{2})

SubNeg

$(II):$ $a - (- b) = a + b$

{y = - 2 x + 6 y = \frac{4}{2} x + \frac{2 0}{2}

CalcQuot

$(II):$ Calculate quotient

{y = - 2 x + 6 y = 2 x + 10

Now that both linear equations are expressed in slope-intercept form, their slopes and

y -

intercepts can be used to draw the lines on the same coordinate plane.

The lines intersect at exactly one point.

Since the system has one or more solutions, it is a consistent system. Furthermore, since it has exactly one solution, it is also an independent system.

b Again, the system of equations will be solved graphically. To do so, both equations will be rewritten in slope-intercept form.

{x + y = 3 4 x + 4 y = 12 (I) (II)

▼

(I), (II):

Write in slope-intercept form

SubEqn

$(I):$ $LHS - x = RHS - x$

{y = - x + 3 4 x + 4 y = 12

SubEqn

$(II):$ $LHS - 4 x = RHS - 4 x$

{y = - x + 3 4 y = - 4 x + 12

DivEqn

$(II):$ $LHS / 4 = RHS / 4$

{y = - x + 3 y = \frac{- 4 x + 1 2}{4}

WriteSumFrac

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

{y = - x + 3 y = \frac{- 4 x}{4} + \frac{1 2}{4}

MoveNegNumToFrac

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

{y = - x + 3 y = - \frac{4 x}{4} + \frac{1 2}{4}

MovePartNumRight

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

{y = - x + 3 y = - \frac{4}{4} x + \frac{1 2}{4}

CalcQuot

$(II):$ Calculate quotient

{y = - x + 3 y = - 1 x + 3

IdPropMult

$(II):$ Identity Property of Multiplication

{y = - x + 3 y = - x + 3

Note that both equations are simplified to the same equation. It can be graphed by using its slope and its

y -

intercept.

The lines overlap each other. Therefore, they intersect at infinitely many points. Since the system has one or more solutions, it is a consistent system. However, since it does not have exactly one solution, it is not an independent system.

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Catch-Up and Review

Basketball and Soccer

Lines on the Same Plane

System of Equations

Equation In Two Variables

System of Equations

Solving a System of Linear Equations Graphically

Modeling With a System of Linear Equations

Hint

Solution

Finding the Solution to a System of Equations by Graphing

Consistent and Independent Systems

Consistent System

Independent System

Finding the Solution and Classifying a System of Equations

Hint

Solution

Dependent and Inconsistent Systems

Dependent System

Inconsistent System

Finding the Solution and Classifying a System of Equations

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