. // 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 Combination Method
Another method to solve a system of linear equations is by using the linear combination method. The linear combination method uses three main steps in order to solve the system.

The Linear Combination Method

Step 1

Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables.

Step 2

Add the revised equations from Step 1. Combining like terms will eliminate one of the variables. Solve for the remaining variable.

Step 3

Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable.

 

Example

Solve this system of linear equations using the linear combination method.
2X - 4Y = 13 Equation 1
4X -5Y = 8 Equation 2

 

Answer  

Step 1

Multiply the first equation by -2 so that the X-coefficients differ only in sign.

2X - 4Y = 13 [multiply by -2] -4X + 8Y = -26
4X -5Y = 8 4X -5Y = 8
Step 2
Add the revised equations and solve for Y.
3Y = -18
Y = -6
Step 3 Substitute the value of Y into one of the original equations and solve for X.

2X - 4Y = 13 Write Equation 1.
2X - 4(-6) = 13 Subsitute -6 for Y.
2X + 24 = 13 Simplify.
X = -11/2 Solve for X.

 

Problem

Okay, now it is your turn to try one. A Diamond Star departs Raleigh the same time a Canadian Regional Jet departs Atlanta both heading towards Boston, which is 1,000 miles away for the CRJ, and 600 miles away for the Star. If the Diamond cuises at 200 mph, and the CRJ cruises at 600 mph, how far from Boston will they be when they meet?

 

Answer

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

-[Speed of the Star] x [Time] + [Original Distance from Boston] = [Distance from Boston]
-[Speed of the CRJ] x [Time] + [Original Distance from Boston] = [Distance from Boston]

Now change the values to make an algebraic model.

Speed of the Star = 200 (mph)
Time = X (hours)
Original Distance from Boston = 600
Distance from Boston = Y
Speed of the CRJ = 600 (mph)
Time = X (hours)
Original Distance from Boston = 100
Distance from Boston = Y

-200X + 600 = Y
-600X + 1000 = Y


Now solve the system of equations using the substitution method.

The two aircraft met 400 miles from Boston.

Step 1

Multiply the first equation by -3 so that the X-coefficients differ only in sign.

-200X + 600 = Y [Multiply by -3] -600X + -1800 = -3Y
-600X + 1000 = Y -600X + 1000 = Y
Step 2
Add the revised equations and solve for Y.
-800 = -2Y
Y = 400
Step 3 Substitute the value of Y into one of the original equations and solve for X.

-200X + 600 = Y Write Equation 1.
-200X + 600 = 400 Subsitute -6 for Y.
-200X = -200 Simplify.
X = 1 Solve for X.
 

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Combination Method

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