Solve a pair of simultaneous linear equations

Equations

\(\displaystyle \begin{gathered}y=2x+1\\y=4-x\end{gathered}\)

A proposed pair is (2,5). Does it solve both equations?

Predict the number of common points if the second line has the same gradient as the first but a different vertical intercept

Line A is y=x+1 and line B is y=mx+1+c. Try m=1,c=2; then m=1,c=0; then m=−1,c=0. Compare parallel, overlapping and crossing traces

When do these two equations describe the same line?

Equations

\(\displaystyle \begin{gathered}y=x+1\\y=x+3\end{gathered}\)

How many pairs satisfy both equations shown?

Equations

\(\displaystyle \begin{gathered}y=x+1\\2y=2x+2\end{gathered}\)

The traces overlap. Which statement describes the complete solution set?

Connect the graph to the algebra

When each of two equations describes a straight line in the coordinate plane, there are three possibilities

1. 1One shared point Nonparallel lines meet once. Find both coordinates and check them in both original equations
2. 2No shared point Distinct parallel lines produce a contradiction when one equation is substituted into the other \(\displaystyle x+1=x+3\Rightarrow1=3\)
3. 3Infinitely many shared points Equivalent equations produce an identity. Their solution set is one common line, not the whole plane \(\displaystyle 2(x+1)=2x+2\)
4. 4Prove equivalence Scale every coefficient and constant by the same nonzero number. Matching only the variable coefficients is insufficient

Equations

\(\displaystyle \begin{gathered}2x-y=3\\6x-3y=9\end{gathered}\)

How many solutions does this pair have?

Equations

\(\displaystyle \begin{gathered}2x-y=3\\6x-3y=10\end{gathered}\)

How many solutions does this pair have?

Two straight lines pass through (0,1). Does that prove the lines coincide?

Equations

\(\displaystyle \begin{gathered}x=-2\\x=3\end{gathered}\)

How many common points do these vertical lines have?

First equation

\(\displaystyle y=3x+2\)

Second equation

\(\displaystyle x+2y=18\)

Which equation correctly replaces the second unknown?

Keep the replacement expression in brackets before expanding

Substitute the whole expression

A simultaneous solution makes both equations true. Use an equation with an unknown already isolated to replace that unknown in the other equation. Then solve the equation with one unknown

1. 1Replace the isolated unknown Keep the replacement in brackets because the second equation multiplies it \(\displaystyle x+2(2x+4)=23\)
2. 2Expand and collect Multiply every term inside the bracket, then collect like terms \(\displaystyle \begin{aligned}x+4x+8&=23\\5x+8&=23\end{aligned}\)
3. 3Solve for the first unknown Subtract the constant from both sides, then divide by the coefficient \(\displaystyle \begin{aligned}5x&=15\\x&=3\end{aligned}\)
4. 4Find the other unknown Substitute the value into the equation with the isolated unknown \(\displaystyle \begin{aligned}y&=2(3)+4\\y&=10\end{aligned}\)
5. 5Check both original equations Both equations must be true for the same pair \(\displaystyle \begin{gathered}10=2(3)+4\\3+2(10)=23\\(x,y)=(3,10)\end{gathered}\)
6. 6When both equations have the same subject Set the two expressions equal. This is also substitution. Then solve and find the other unknown as before \(\displaystyle \begin{gathered}y=2x+5\\y=11-x\\2x+5=11-x\end{gathered}\)

First equation

\(\displaystyle y=3x-2\)

Second equation

\(\displaystyle 2x+y=23\)

Find the first unknown in this simultaneous pair

Replace the second unknown in the second equation with the whole expression from the first equation

First equation

\(\displaystyle y=9+\frac{x}{2}\)

Second equation

\(\displaystyle y=1-\frac{3x}{2}\)

Find the second unknown in this simultaneous pair

Both equations already have the same subject. Equate the two expressions before finding the requested value

First equation

\(\displaystyle y=4x-3\)

Second equation

\(\displaystyle 2x-y=7\)

Which line contains the first error?

Check how each term changes when the bracket is removed