In Exercises 5–18, solve each system by the substitution method. y = (1/3)x + 2/3 y = (5/7)x - 2

Verified step by step guidance

1. Step 1: Since both equations are equal to y, set the right-hand sides of the equations equal to each other: $\frac{1}{3}x\; +\frac{2}{3}=\frac{5}{7}x\; -\; 2$.
2. Step 2: To solve for x, first eliminate the fractions by finding a common denominator or multiply both sides of the equation by the least common multiple of the denominators (which is 21) to clear the fractions.
3. Step 3: After clearing the fractions, simplify the resulting equation and isolate the variable x on one side.
4. Step 4: Solve the simplified linear equation to find the value of x.
5. Step 5: Substitute the value of x back into either original equation (for example, $y\; =\frac{1}{3}x\; +\frac{2}{3}$) to find the corresponding value of y.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution is the set of values that satisfy all equations simultaneously. Graphically, this corresponds to the point(s) where the lines intersect.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve.

Slope-Intercept Form

Equations in slope-intercept form are written as y = mx + b, where m is the slope and b is the y-intercept. Understanding this form helps in interpreting the equations and solving systems by substitution or graphing.