A linear equation in one variable can be solved by a chain of justified steps: distribute, combine like terms, clear fractions, move variable terms to one side, and divide by the coefficient. Each step uses a property of equality, and the answer is checked in the original equation. Literal equations use the same steps for formulas.
Key facts:
- Standards A-REI.A.1, A-REI.B.3, A-CED.A.1 and A-CED.A.4 cover justified solving, multi-step equations, writing equations and rearranging formulas.
- Every step must be backed by a property: equality properties, the distributive property, or combining like terms.
- Multiply by the least common denominator to clear fractions before solving.
- If the variable cancels, a false statement means no solution and a true statement means infinitely many solutions.
- To solve a formula for one letter, treat the other letters as numbers: for A = (1/2)bh, h = 2A/b.
What does it mean to justify a step in solving an equation?
To solve a linear equation, you rewrite it as a chain of simpler equations that all have the same solution. Each move must be backed by a reason, such as a property of equality or the distributive property. If every step has a reason, the final answer is guaranteed to satisfy the original equation.
Addition and subtraction properties of equality: adding or subtracting the same number on both sides keeps the equation true.
Multiplication and division properties of equality: multiplying or dividing both sides by the same nonzero number keeps it true.
Distributive property: \(a(b+c)=ab+ac\). Combining like terms adds the coefficients of matching variable terms.
How do you solve a multi-step linear equation?
- Clear parentheses with the distributive property and combine like terms on each side.
- Clear fractions or decimals by multiplying every term by the least common denominator or a power of 10.
- Collect the variable terms on one side and the constants on the other, using inverse operations.
- Divide by the coefficient, then check the answer in the original equation.
Worked example. Solve \(3(x+2)=2x+11\). Distribute: \(3x+6=2x+11\). Subtract \(2x\): \(x+6=11\). Subtract 6: \(x=5\). Check: \(3(7)=21\) and \(2(5)+11=21\).
With fractions. To solve \(\frac{x}{2}+\frac{x}{3}=10\), multiply every term by 6 to get \(3x+2x=60\), so \(x=12\).
Can a linear equation have no solution or infinitely many?
Yes. Simplify until the variable cancels. A false statement such as \(2=5\) means no solution. A true statement such as \(2=2\) means infinitely many solutions: every real number works.
How does a graph show the solution of an equation?
Graph the left side and the right side as two lines. The \(x\)-coordinate where they cross is the solution. For \(2x+1=x+4\) the lines meet at \((3,7)\), so \(x=3\).
How do you write an equation from a word problem?
Name the unknown, then translate each phrase. A taxi charges a \(\$3.50\) fee plus \(\$2.25\) per mile. If the ride costs \(\$17.00\), then \(3.50+2.25m=17\), so \(m=6\) miles.
What is a literal equation?
A literal equation is a formula with several letters. To solve for one letter, treat the others as numbers and use inverse operations. For \(A=\frac{1}{2}bh\), multiply both sides by 2 to get \(2A=bh\), then divide by \(b\): \(h=\frac{2A}{b}\).
Linear Equations and Literal Equations: frequently asked questions
How do I solve an equation with variables on both sides?
Move the variable terms to one side by adding or subtracting the same term on both sides. For \(7x-3=5x+11\), subtract \(5x\) to get \(2x-3=11\), add 3, then divide by 2 to find \(x=7\).
How do I get rid of fractions in an equation?
Multiply every term on both sides by the least common denominator. In \(\frac{x}{2}+\frac{x}{3}=10\) the denominator is 6, which gives \(3x+2x=60\) and \(x=12\).
What does it mean when an equation has no solution?
It means no number makes both sides equal. After simplifying, you get a false statement such as \(6=5\). The two sides, graphed as lines, are parallel and never meet.
What is the difference between an identity and an equation with one solution?
An identity such as \(3(x+2)=3x+6\) is true for every value of \(x\), so it has infinitely many solutions. An equation with one solution is true for exactly one value.
How do I solve a formula for a different variable?
Use the same inverse operations as for numbers, in reverse order of operations. To solve \(P=2l+2w\) for \(w\), subtract \(2l\) and divide by 2, giving \(w=\frac{P-2l}{2}\).
See every chapter on the Algebra 1 math page.