In Lesson 13–2, you learned to solve systems of equations by graphing. But sometimes the exact coordinates of the point where lines intersect cannot be easily determined from a graph. The solution of a system can also be found by using an algebraic method called substitution.
Use substitution to solve each system of equations.
1.
y = 2x
3x
+ y = 5
Alternative Solutions:
The
first equation tells you that y is equal to 2x. So, substitute 2x
for y in the second equation. Then solve for x.
Now substitute 1 for x in
either equation and solve for y. Choose the equation that is easier to
solve.
The solution of this system of equations is
(1, 2). You can see from the graph that the solution is correct. You can also check by substituting (1, 2) into each of the original
equations.
2.
x + y = 1
x = y + 6
Alternative Solutions:
Substitute
y 6 for x in the first equation. Then solve for y.
Now substitute 
for y in either
equation and solve for x.
The solution of this system of equations is 
. Check by substituting 
into each of the original equations.
Example
3.
x – 3y = 3
2x
– y = 11
Alternative Solutions:
Solve
the first equation for x since the coefficient of x is 1.
x – 3y = 3 → x =
3 + 3y
The
solution is (6, 1).
In Lesson 13–2, you learned how to tell whether a system has
one solution, no solution, or infinitely many solutions by looking at the
graph. You can also determine this information algebraically.
Example
Use substitution to solve each system of equations.
4. y = 4x + 1
4x
– y = 7
Alternative Solutions:
Find
the value of x by substituting 4x + 1 for y in the second
equation.
The
statement –1 = 7 is false. This means that there are no ordered pairs that are solutions
to both equations. Compare the slope-intercept forms of the equations, y =
4x + 1 and y = 4x – 7. Notice that the graphs of these
equations have the same slope but different y-intercepts. Thus, the lines
are parallel, and the system has no solution.
5.
x = 3 – 2y
2x
+ 4y = 6
Alternative Solutions:
The
statement 6 = 6 is true. This means that an ordered pair for any point on
either line is a solution to both equations. The system has infinitely many
solutions.
Systems of equations can be used to solve mixture problems.
Metals
Link
6. A certain metal alloy is 25% copper.
Another metal alloy is 50% copper. How much of each alloy should be used to
make 1000 grams of a metal alloy that is 45% copper?
Alternative Solutions:
Sumber
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