. // I recommend you leave this here as otherwise when you resize NS4's width menus are hidden. var popOldWidth = window.innerWidth; nsResizeHandler = new Function('if (popOldWidth != window.innerWidth) location.reload()'); // This is a quick snippet that captures all clicks on the document and hides the menus // every time you click. Use if you want. if (isNS4) document.captureEvents(Event.CLICK); document.onclick = clickHandle; function clickHandle(evt) { if (isNS4) document.routeEvent(evt); hideAllBut(0); } // This is just the moving command for the example. function moveRoot() { with(menu[0][0].ref) left = ((parseInt(left) < 100) ? 100 : 5); } // End -->

Solving Linear Systems By the Substitution Method
Another method to solve a system of linear equations is by using the substitution method. The substitution method uses three main steps in order to solve the system.

The Substitution Method

Step 1

Solve one of the equations for one of its variables

Step 2

Substitute the expression from Step 1 into the other equation and solve for the other variables.

Step 3

Substitute the value from Step 2 into the revised equation from Step 1 and solve.

 

Example

Solve this system of linear equations using the substitution method.
3X + 4Y = -4 Equation 1
X + 2Y = 2 Equation 2

 

Answer
  

Step 1

Solve Equation 2 for X

X + 2Y = 2 Write Equation 2.
X = -2y + 2 Rewrite Equation 2 so that one variable is by itself.
Step 2 Substitute the expression for X into Equation 1 and solve for Y.

3X + 4Y = -4 Write Equation 1.
3(-2Y + 2) + 4Y = -4 Subsitute -2Y + 2 for X because X is equal to this.
Y = 5 Solve for Y.
Step 3 Substitute the value of Y into the revised Equation 2and solve for X.

X = -2y + 2 Write revised Equation 2.
X = -2(5) + 2 Subsitute -2Y + 2 for X because X is equal to this.
X = -8 Simplify.

 

Problem

Okay, now it is your turn to try one. A Diamond Eclipse is departing at the same time a Cessna 182 is descending through 3,600 feet. If the Eclipse is climbing at 500 feet per minute and the Cessna is descending at 700 feet per minute, in how many minutes will they be at the same altitude?

 

Answer

Start the problem by creating a verbal model for the two equations.

[Cessna's Altitude] = [Descent Rate] X [Time] + [Starting Altitude]

[Diamond's Altitude] = [Climb Rate] X [Time]

Now change the values to make an algebraic model.

Cessna's Altitude = Y (feet)
Descent Rate = -500 (feet per minute)
Time = X (minutes)
Starting Altitude = 3600 (feet)
Diamond's Altitude = Y (feet)
Climb Rate= 700 (feet)
Time = X (minutes)

Y = -500X + 3600
Y = 700X

Now solve the system of equations using the substitution method.

Step 1

Equation 1 already has a variable isolated so we can move on to step two.
Step 2 Substitute the expression for Y into Equation 2 and solve for X.

Y = 700X Write Equation 2.
-500X + 3600 = 700X Subsitute 500X + 3600 for Y because Y is equal to this.
X= 3 Solve for X. The aircraft crossed paths three minutes after the Diamond began its takeoff, and the Cessna began its descent. This gives you the answer to the problem, but if you wish to see the altitude that they met at, continue at step three.
Step 3 Substitute the value of X into Equation 2 and solve for Y.

Y = 700X Write Equation 2.
Y = 700(3) Subsitute 3 for X.
Y = 2100 Simplify. The aircraft were at 2100 feet when they crossed paths.
 

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Copyright © 2004 by Aviation Equations Inc. by John Tarleton
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