Solving System of Equations Algebraically - Lines on the Coordinate Plane (Pre-Algebra)

a In order to solve the system of equations by graphing, both equations should be written in slope-intercept form.

y = m x + b

Rewrite both equations until they match this form. Notice that the first equation is already almost in this form — all that is left is to subtract

4 m

from both sides.

p + 4 m - 4 m = 42 - 4 m ⇕ p = - 4 m + 42

Rewrite the second equation similarly.

8 p - 5 m = 3

▼

Write in slope-intercept form

AddEqn

$LHS + 5 m = RHS + 5 m$

8 p = 3 + 5 m

DivEqn

$LHS / 8 = RHS / 8$

p = \frac{3 + 5 m}{8}

WriteSumFrac

Write as a sum of fractions

p = \frac{3}{8} + \frac{5 m}{8}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

p = \frac{3}{8} + \frac{5}{8} m

CommutativePropAdd

Commutative Property of Addition

p = \frac{5}{8} m + \frac{3}{8}

Now, graph both equations on the same coordinate plane. To graph the first equation, start by plotting the

y -

intercept of

42 .

Next, use the slope of

- 4

to move

1

unit to the right and

4

units down, or

2

units to the right and

2 \cdot 4 = 8

units down, to plot the second point.

Draw a line through the two plotted points to get the graph of the first equation.

The second equation can be graphed by following the same process.

The solution of the system of equations is represented by the point of intersection of the lines. If the point of intersection lies on lattice lines or their intersections, the exact solution will be determined. Otherwise, only an estimate of the solution might be found.

The lines intersect at $(9, 6) .$ Therefore, $m = 9$ and $p = 6,$ which indicates that Vincenzo spent $9$ minutes spacewalking and installed $6$ parts on the spaceship.

b The system of equations will now be solved by using the Substitution Method. Start by isolating the variable

p

and substituting the corresponding expression into the second equation.

{p + 4 m = 42 8 p - 5 m = 3 (I) (II)

SubEqn

$(I):$ $LHS - 4 m = RHS - 4 m$

{p = 42 - 4 m 8 p - 5 m = 3

Substitute

$(II):$ $p = 42 - 4 m$

{p = 42 - 4 m 8 (42 - 4 m) - 5 m = 3

Distr

$(II):$ Distribute $8$

{p = 42 - 4 m 336 - 32 m - 5 m = 3

SubTerm

$(II):$ Subtract term

{p = 42 - 4 m 336 - 37 m = 3

SubEqn

$(II):$ $LHS - 336 = RHS - 336$

{p = 42 - 4 m - 37 m = - 333

DivEqn

$(II):$ $LHS / (- 37) = RHS / (- 37)$

{p = 42 - 4 m m = 9

The value of

m

is found to be

9 .

Now it can be substituted in either of the original equations. Notice that

p

is already isolated in the first equation, so it might be convenient to substitute the value of

m

into this equation and evaluate

p .

{p = 42 - 4 m m = 9

Substitute

$(I):$ $m = 9$

{p = 42 - 4 (9) m = 9

Multiply

$(I):$ Multiply

{p = 42 - 36 m = 9

SubTerm

$(I):$ Subtract term

{p = 6 m = 9

The solution to the system of equations is

p = 6

and

m = 9 .

c Both methods of solving the system of equations gave the same solution. Therefore, both methods of solving are correct.

Graphing Method ↘ (9, 6) Substitution Method ↙

However, in this case, the Substitution Method can be more convenient because it is shorter and gives the exact solution. By comparison, the graphing method requires the equations to be in slope-intercept form and does not always result in finding the exact solution.

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Catch-Up and Review

Books and Movies About Space

Substitution Method of Solving a System of Equations

Substitution Method

First Time Spacewalking

Answer

Hint

Solution

Distances Between Planets

Answer

Hint

Solution

Elimination Method

Pit Stop on Exosia

Answer

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